An Introduction to Derivative Securities, Financial Markets, and Risk Management: 2nd Edition [2 ed.] 1944659552, 9781944659554

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Copyright © 2018. World Scientific Publishing Company Pte. Limited. All rights reserved.

AN INTRODUCTION TO

Derivative Securities, Financial Markets, and Risk Management ROBERT JARROW

Second Edition

ARKADEV CHATTERJEA

World Scientific

AN INTRODUCTION TO

Derivative Securities, Financial Markets, and Risk Management Second Edition

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AN INTRODUCTION TO

Derivative Securities, Financial Markets, and Risk Management Second Edition

ROBERT A. JARROW Cornell University

ARKADEV CHATTERJEA Indiana University Bloomington

World Scientific NEW JERSEY

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LONDON

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SINGAPORE

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BEIJING

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SHANGHAI

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HONG KONG

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TAIPEI

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CHENNAI

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TOKYO

Published by World Scientific Publishing Co. Inc. 27 Warren Street, Suite 401-402, Hackensack, NJ 07601, USA Head office: 5 Toh Tuck Link, Singapore 596224 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

Library of Congress Cataloging-in-Publication Data Names: Jarrow, Robert A., author. | Chatterjea, Arkadev, author. Title: An introduction to derivative securities, financial markets, and risk management / Robert Jarrow (Cornell University), Arkadev Chatterjea (Indiana University). Description: 2nd edition. | New Jersey : World Scientific, [2019] | Includes bibliographical references and index. Identifiers: LCCN 2018022534 | ISBN 9781944659554 Subjects: LCSH: Derivative securities. | Financial institutions. | Capital market. | Risk management. Classification: LCC HG6024.A3 J3747 2018 | DDC 332.64/57--dc23 LC record available at https://lccn.loc.gov/2018022534

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

Copyright © 2019 by Robert Jarrow and Arkadev Chatterjea All rights reserved.

For any available supplementary material, please visit https://www.worldscientific.com/worldscibooks/10.1142/Y0018#t=suppl

Desk Editors: Karimah Samsudin and Shreya Gopi

Typeset by diacriTech

Printed in Singapore

Bob: To my wife Gail for her patience and understanding. Arka: To my wife Sudeshna for her cheerful and steadfast support, and to our daughters Rushtri, Tvisha, and Roudra (all younger than the book!), who also cheerfully and proudly supported my writing.

About the Authors Robert A. Jarrow is the Ronald P. and Susan E. Lynch Professor of Investment Management at the Samuel Curtis Johnson Graduate School of Management, Cornell SC Johnson College of Business. He is among the most distinguished finance scholars of his generation. Jarrow has done research in nearly all areas of derivatives pricing. He is the co-developer of two widely used pricing models in finance, the Heath–Jarrow– Morton (HJM) model for pricing interest-rate derivatives and the reduced form model for pricing securities with credit risk. He is the author of more than 200 academic publications, seven books including Option Pricing (with Andrew Rudd, 1983), Modelling Fixed Income Securities and Interest Rate Options (1996), and Derivative Securities (with Stuart Turnbull, 2000), and several edited volumes. Arkadev Chatterjea is a Visiting Professor of Finance, Kelley School of Business, Indiana University Bloomington. He is also a Research Fellow at UNC Chapel Hill and a Visiting Fellow at CHERI, Cornell University. He did his Ph.D. at Cornell, where he was a student of Jarrow. Earlier, he was a professor of finance at the Indian Institute of Management Calcutta. A winner of research and teaching awards in the USA, Chatterjea has taught derivatives at the above universities and at other institutions including CU Boulder, the Helsinki School, Hong Kong UST, and IIM Ahmedabad. Photo by Mr. Kallol Nath.

Brief Contents

About the Authors VI Preface to Second Edition XXVII Preface to First Edition XXXI

17 Single-Period Binomial Model 367 18 Multiperiod Binomial Model 392 19 The Black–Scholes–Merton Model 422

PART I Introduction 1 Derivatives and Risk Management 2 2 Interest Rates 22

20 Using the Black–Scholes–Merton Model 462

3 Stocks 53

PART IV Interest Rate Derivatives

4 Forwards and Futures 72

21 Yields and Forward Rates 496

5 Options 92

22 Interest Rate Swaps 538

6 Arbitrage and Trading 112

23 Single-Period Binomial Heath–Jarrow–Morton Model 558

7 Financial Engineering and Swaps 129

24 Multiperiod Binomial HJM Model 592

PART II Forwards and Futures 8 Forwards and Futures Markets 154 9 Futures Trading 172 10 Futures Regulations 192 11 The Cost-of-Carry Model 209

25 The Heath–Jarrow–Morton Libor Model 618 26 Risk Management Models 655 Appendix A: Mathematics and Statistics 684

12 The Extended Cost-of-Carry Model 229

Glossary 693

13 Futures Hedging 256

Notation 710

References 704

Additional Sources and Websites 712

PART III Options 14 Options Markets and Trading 288

Books on Derivatives and Risk Management 715

15 Option Trading Strategies 310

Name-Index 718

16 Option Relations 336

Subject-Index 720

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Contents

About the Authors VI Preface to Second Edition XXVII Preface to First Edition XXXI

PA R T I Introduction CHAPTER 1 Derivatives and Risk Management

.....................

2

1.1 Introduction 3 1.2 Financial Innovation 5 Expanding Derivatives Markets 5 Two Economic Motives 7 1.3 Traded Derivative Securities 7 EXTENSION 1.1: The Influence of Regulations, Taxes, and Transaction Costs on Financial Innovation 8 Diverse Views on Derivatives 9 Applications and Uses of Derivatives 10 A Quest for Better Models 12 1.4 Defining, Measuring, and Managing Risk 12 1.5 The Regulator’s Classification of Risk 12 1.6 Portfolio Risk Management 14 1.7 Corporate Financial Risk Management 14 Risks That Businesses Face 14 Nonhedged Risks 16 Risk Management in a Blue Chip Company 16 1.8 Risk Management Perspectives in This Book 17 1.9 Summary 18 1.10 Cases 19 1.11 Questions and Problems 19

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CHAPTER 2 Interest Rates

.............................................................. 22

2.1 Introduction 23 2.2 Rate of Return 24 2.3 Basic Interest Rates: Simple, Compound, and Continuously Compounded 25 EXTENSION 2.1: Conventions and Rules for Rounding, Reporting Numbers, and Measuring Time 26 2.4 Discounting (PV) and Compounding (FV): Moving Money across Time 31 EXTENSION 2.2: Moving Multiple Cash Flows across Time 34 2.5 US Treasury Securities 37 2.6 US Federal Debt Auction Markets 38 2.7 Different Ways of Investing in Treasury Securities 40 The Treasury Auction and Its Associated Markets 40 The Repo and the Reverse Repo Market 41 EXTENSION 2.3: A Repurchase Agreement 41 Interest Rate Derivatives 43 2.8 Treasury Bills, Notes, Bonds, and STRIPS 43 2.9 Libor versus a Libor Rate Index 47 2.10 Summary 48 2.11 Cases 50 2.12 Questions and Problems 50

CHAPTER 3 Stocks

............................................................................. 53

3.1 Introduction 54 3.2 Primary and Secondary Markets, Exchanges, and Over-the-Counter Markets 54 3.3 Brokers, Dealers, and Traders in Securities Markets 56 3.4 Automation of Trading 58 3.5 The Three-Step Process of Transacting Exchange-Traded Securities 58 3.6 Buying and Selling Stocks 59 Trading at the New York Stock Exchange 60 Over-the-Counter Trading 61 Alternative Trading Systems: Dark Pools and Electronic Communications Networks 61 3.7 Dollar Dividends and Dividend Yields 62 3.8 Short Selling Stocks 64

CONTENTS

3.9 Margin–-Security Deposits That Facilitate Trading 66 EXTENSION 3.1: Margin and Stock Trading 67 3.10 Summary 68 3.11 Cases 70 3.12 Questions and Problems 70

CHAPTER 4 Forwards and Futures

.............................................. 72

4.1 Introduction 73 4.2 Forward Contracts 73 4.3 The Over-the-Counter Market for Trading Forwards 78 4.4 Futures Contracts 81 4.5 Exchange Trading of a Futures Contract 83 EXTENSION 4.1: Futures Exchanges in China and India 86 4.6 Hedging with Forwards and Futures 88 4.7 Summary 89 4.8 Cases 90 4.9 Questions and Problems 90

CHAPTER 5 Options

.......................................................................... 92

5.1 Introduction 93 5.2 Options 93 5.3 Call Options 95 Call Payoffs and Profit Diagrams 95 The Call’s Intrinsic and Time Values 98 Price Bounds for American Calls 99 5.4 Put Options 100 Put Payoffs and Profit Diagrams 100 The Put’s Intrinsic and Time Values 102 Price Bounds for American Puts 103 5.5 Exchange-Traded Options 105 5.6 Longs and Shorts in Different Markets 107 5.7 Order Placement Strategies 107 5.8 Summary 108 5.9 Cases 109 5.10 Questions and Problems 109

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CHAPTER 6 Arbitrage and Trading

........................................... 112

6.1 Introduction 113 6.2 The Concept of Arbitrage 113 6.3 The No-Arbitrage Principle for Derivative Pricing 117 The Law of One Price 117 Nothing Comes from Nothing 117 6.4 Efficient Markets 118 6.5 In Pursuit of Arbitrage Opportunities 119 The Closed-End Fund Puzzle 119 Spread Trading 120 Index Arbitrage 121 EXTENSION 6.1: Stock Indexes 121 EXTENSION 6.2: Index Arbitrage 123 6.6 Illegal Arbitrage Opportunities 124 6.7 Summary 126 6.8 Cases 127 6.9 Questions and Problems 127

CHAPTER 7 Financial Engineering and Swaps

..................... 129

7.1 Introduction 130 7.2 The Build and Break Approach 130 7.3 Financial Engineering 131 Cash Flows versus Asset Values 131 Examples 132 7.4 An Introduction to Swaps 135 7.5 Applications and Uses of Swaps 135 7.6 Types of Swaps 136 Interest Rate Swaps 136 Forex Swaps 138 Currency Swaps 139 EXTENSION 7.1: Valuing Fixed-for-Fixed Currency Swaps 142 Commodity and Equity Swaps 146 Credit Default Swaps 147 7.7 Summary 148 7.8 Cases 149 7.9 Questions and Problems 149

CONTENTS

PA R T II Forwards and Futures CHAPTER 8 Forwards and Futures Markets

....................... 154

8.1 Introduction 155 8.2 Applications and Uses of Forwards and Futures 155 8.3 A Brief History of Forwards and Futures 156 Early Trading of Forward and Futures-Type Contracts 156 US Futures Exchanges and the Evolution of the Modern Futures Contract, 1848–1926 157 Recent Developments, 1970 Onward 159 8.4 Futures Contract Features and Price Quotes 161 Commodity and Financial Futures Contracts 161 The Gold Futures Contract 161 Gold Futures Price Quotes 164 The Exchange and Clearinghouse 166 8.5 Commodity Price Indexes 167 EXTENSION 8.1: Doomsters and Boomsters 168 8.6 Summary 169 8.7 Cases 170 8.8 Questions and Problems 170

CHAPTER 9 Futures Trading

...................................................... 172

9.1 Introduction 173 9.2 Brokers, Dealers, and the Futures Industry 173 9.3 Trading Futures 174 9.4 Margin Accounts and Daily Settlement 176 Daily Settlement 176 9.5 Futures and Forward Price Relations 180 Equality of Futures and Spot Price at Maturity 183 Equality of Forward and Futures Prices before Maturity 183 Convergence of the Basis 185 9.6 Trading Spreads 185 9.7 Summary 188 9.8 Cases 189 9.9 Questions and Problems 189

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CHAPTER 10 Futures Regulations

............................................. 192

10.1 Introduction 193 10.2 Which Markets Have Futures? 193 Price Discovery 194 Hedging 194 Speculation 195 EXTENSION 10.1: When Sellers Have Market Power 195 10.3 Regulation of US Futures Markets 196 Major Regulatory Acts 196 The CFTC, the NFA, and the Regulatory Role of Exchanges 199 Enforcement Actions by the CFTC and the NFA 201 10.4 Manipulation in Futures Markets 202 The Hunts Silver Case 202 Salomon Brothers and US Treasury Securities 203 Banging the Close 204 10.5 Managing Commodity Markets 205 10.6 Summary 206 10.7 Cases 206 10.8 Questions and Problems 207

CHAPTER 11 The Cost-of-Carry Model

.................................. 209

11.1 Introduction 210 11.2 A Cost-of-Carry Example 210 11.3 The Assumptions 214 11.4 The Cost-of-Carry Model 216 The Model Setup 216 Using the Law of One Price 217 Using Nothing Comes from Nothing 217 Different Methods for Computing Interest 218 The Arbitrage Table Approach 220 11.5 Valuing a Forward Contract at Intermediate Dates 221 11.6 Linking Forward Prices of Different Maturities 224 11.7 Summary 225 11.8 Cases 226 11.9 Questions and Problems 226

CONTENTS

CHAPTER 12 The Extended Cost-of-Carry Model

............. 229

12.1 Introduction 230 12.2 A Family of Forward Pricing Models 230 12.3 Forwards on Dividend-Paying Stocks 232 The Cost-of-Carry Model with Dollar Dividends 232 The Cost-of-Carry Model with a Dividend Yield 235 A Synthetic Index 236 The Foreign Currency Forward Price 239 12.4 Extended Cost-of-Carry Models 243 12.5 Backwardation, Contango, Normal Backwardation, and Normal Contango 246 12.6 Market Imperfections 249 12.7 Summary 252 12.8 Cases 253 12.9 Questions and Problems 253

CHAPTER 13 Futures Hedging

................................................... 256

13.1 Introduction 257 13.2 To Hedge or Not to Hedge 257 Should the Firm or the Individual Hedge? 257 The Costs and Benefits of Corporate Hedging 259 EXTENSION 13.1: Airlines and Fuel Price Risk 260 EXTENSION 13.2: A Hedged Firm Capturing a Tax Loss 262 13.3 Hedging with Futures 264 Perfect and Imperfect Hedges 264 Basis Risk 264 Guidelines for Futures Hedging 267 13.4 Risk-Minimization Hedging 268 The Mean-Variance Approach 268 Limitations of Risk-Minimization Hedging 272 13.5 Futures versus Forward Hedging 272 13.6 Spreadsheet Applications: Computing h 273 13.7 Summary 275 13.8 Appendix 276 Deriving the Minimum-Variance Hedge Ratio (h) 276 Computing the Minimum-Variance Hedge Ratio (h) 277

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Statistical Approach 277 Econometric Approach: A Linear Regression Model 281 13.9 Cases 281 13.10 Questions and Problems 282

PA R T III Options CHAPTER 14 Options Markets and Trading

......................... 288

14.1 Introduction 289 14.2 Exchange-Traded Options 289 14.3 A History of Options 290 Early Trading of Options 291 The Year 1973: The Watershed Year and After 293 EXTENSION 14.1: Options on Futures 295 14.4 Option Contract Features 296 Maturity Dates 297 Strike Prices 297 Margin Requirements and Position Limits 298 Dividends and Stock Splits 299 14.5 Options Trading, Exercising, and the Expiration Process 300 The Trading Process 300 Exercising the Option 300 The Expiration Process 301 14.6 Options Price Quotes 301 14.7 Regulation and Manipulation in Options Markets 305 14.8 Summary 306 14.9 Cases 307 14.10 Questions and Problems 307

CHAPTER 15 Option Trading Strategies 15.1 Introduction 311 15.2 Traders in Options Markets 311 15.3 Profit Diagrams 312 15.4 Options Strategies 313

................................ 310

CONTENTS

Common Options Price Data 313 Naked Trades: The Basic Building Blocks 314 15.5 Hedged Strategies 317 15.6 Spread Strategies 320 EXTENSION 15.1: Collars, Ratio Spreads, Butterflies, and Condors 323 EXTENSION 15.2: Reinsurance Contracts and Call Spreads 327 15.7 Combination Strategies 329 15.8 Summary 331 15.9 Cases 332 15.10 Questions and Problems 332

CHAPTER 16 Option Relations

................................................... 336

16.1 Introduction 337 16.2 A Graphical Approach to Put–Call Parity 337 16.3 Put–Call Parity for European Options 338 16.4 Market Imperfections 342 EXTENSION 16.1: Put–Call Parity in Imperfect Markets 343 EXTENSION 16.2: Dividends and American Options 344 16.5 Options Price Restrictions 346 EXTENSION 16.3: The Superglue Argument 352 16.6 Early Exercise of American Options 357 No Dividends 357 Dividends and Early Exercise 360 16.7 Summary 362 16.8 Cases 364 16.9 Questions and Problems 364

CHAPTER 17 Single-Period Binomial Model 17.1 Introduction 368 17.2 Applications and Uses of the Binomial Model 368 17.3 A Brief History of Options Pricing Models 369 Option Pricing Pre-1973 369 Options Pricing, 1973 and After 370 17.4 An Example 372

........................ 367

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17.5 The Assumptions 374 17.6 The Single-Period Model 376 The No-Arbitrage Principle 376 Building Binomial Trees 377 Arbitrage-Free Trees 378 Stock Prices and Martingales 378 The Pricing Model 379 The Hedge Ratio 380 Risk-Neutral Valuation 381 Actual versus Pseudo-probabilities 382 17.7 Summary 383 17.8 Appendix 384 Proving the No-Arbitrage Argument 384 The Probabilities and Risk Premium 385 17.9 Cases 388 17.10 Questions and Problems 388

CHAPTER 18 Multiperiod Binomial Model

............................ 392

18.1 Introduction 393 18.2 Toward a Multiperiod Binomial Option Pricing Model 393 The Stock Price Evolution 393 Binomial Option Price Data 394 The Stock Price Tree 395 18.3 A Two-Period Binomial Model 396 Backward Induction 396 Option Pricing via Synthetic Construction (Method 1) 397 Repeat, Repeat: Risk-Neutral Pricing (Method 2) 400 One-Step Valuation: Prelude to the Multiperiod Model (Method 3) 401 18.4 The Multiperiod Binomial Option Pricing Model 403 Binomial Coefficients and Pseudo-probabilities 403 Recasting the Two-Period Example in the Multiperiod Framework 403 The n-Period Binomial Option Pricing Model 405 EXTENSION 18.1: Linking the Binomial Model to the Black–Scholes–Merton Model 407 18.5 Extending the Binomial Model 409 Known Dividends 409 Valuing American Options 410

CONTENTS

18.6 Spreadsheet Applications 413 A Two-Period Binomial Example 413 A Sixteen-Period Example 416 18.7 Summary 418 18.8 Cases 419 18.9 Questions and Problems 419

CHAPTER 19 The Black–Scholes–Merton Model

............... 422

19.1 Introduction 423 19.2 Nobel Prize–Winning Works (1973) 423 19.3 The Assumptions 424 EXTENSION 19.1: Bubbles and Option Pricing 424 19.4 The Pricing and Hedging Argument 428 19.5 The Black–Scholes–Merton Formula 429 19.6 Understanding the Black–Scholes–Merton Model 432 Stock Prices and Martingales 432 Risk-Neutral Valuation 433 Actual versus Pseudo-probabilities 433 19.7 The Greeks 434 Interpreting the Greeks 434 Some Road Bumps Ahead 439 EXTENSION 19.2: Market Manipulation and Option Pricing 439 19.8 The Inputs 440 Observable Inputs 441 Volatility: The Elusive Input 441 19.9 Extending the Black–Scholes–Merton Model 444 Adjusting for Dividends 444 Foreign Currency Options 446 Valuing American Options 447 EXTENSION 19.3: Exotic Options 448 19.10 Summary 449 19.11 Appendix 451 Modeling the Stock Price Evolution 451 Continuously Compounded (Logarithmic) Return 451 Introducing Uncertainty 452 The Central Limit Theorem 453

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First Derivation of the Black–Scholes–Merton Formula (as a Limit of the Binomial Model) 454 Second Derivation of the Black–Scholes–Merton Formula (Using Risk-Neutral Valuation) 457 The Probabilities and the Risk Premium 457 Proof of (24) 458 19.12 Cases 459 19.13 Questions and Problems 459

CHAPTER 20 Using the Black–Scholes–Merton Model 20.1 Introduction 463 20.2 Hedging the Greeks 463 Delta Hedging 463 Gamma Hedging 466 20.3 Hedging a portfolio of options 469 20.4 Vega Hedging 470 EXTENSION 20.1: Stochastic Volatility Option Models 473 20.5 Calibration 474 Theoretical and Econometric Models 474 Using Calibration 475 Implied Volatilities 477 EXTENSION 20.2: VIX: The Fear Index 480 20.6 The Black–Scholes–Merton Model: A Postscript 480 20.7 Summary 481 20.8 Appendix 482 The Mathematics of Delta, Gamma, and Vega Hedging 482 A Third Derivation of the Black–Scholes–Merton Formula 486 20.9 Cases 487 20.10 Questions and Problems 488 Overview 490 Data (from the Wall Street Journal, March 3, 2000) 490 The Project 491

... 462

CONTENTS

PA R T IV Interest Rate Derivatives CHAPTER 21 Yields and Forward Rates

.................................. 496

21.1 Introduction 497 EXTENSION 21.1: Home Ownership and Derivatives 498 21.2 Yields 499 Revisiting Zeros 499 Yields and the Yield Curve 500 EXTENSION 21.2: The Expectations Hypothesis 504 21.3 The Traditional Approach 507 Duration 507 Modified Duration Hedging 510 Applications and Limitations 512 21.4 Forward Rates 513 The Definition 514 Understanding Forward Rates 516 Using Forward Rates 517 EXTENSION 21.3: Computing Forward Rates from Coupon Bond Prices 519 21.5 The Basic Interest Rate Derivatives Contracts 520 A Brief History 520 Forward Rate Agreements 521 Interest Rate Futures 524 EXTENSION 21.4: Treasury Futures 529 The Equivalence between Forward and FRA Rates 530 21.6 Summary 531 21.7 Cases 534 21.8 Questions and Problems 535

CHAPTER 22 Interest Rate Swaps

............................................. 538

22.1 Introduction 539 22.2 A Brief History 539 The Introduction of Swap Contracts 540 From a Brokerage to a Dealership Market 541 ISDA and Standardization of Contracts 541

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CONTENTS

22.3 Institutional Features 542 Entering a Swap Contract 542 Documentation 543 Closing a Swap Position 544 22.4 Valuation 544 22.5 Variations of Interest Rate Swaps 549 22.6 Swaps and FRAs 550 Synthesizing Swaps with Eurodollars and FRAs 550 The Yield and Swap Curve 552 EXTENSION 22.1: Computing Forward Rates from Swap Rates 553 22.7 Summary 555 22.8 Cases 555 22.9 Questions and Problems 555

CHAPTER 23 Single-Period Binomial Heath–Jarrow–Morton Model

......................... 558

23.1 Introduction 559 23.2 The Basic Interest Rate Derivatives 560 Uses of Caps, Floors, and Collars 561 23.3 A Brief History of Interest Rate Derivatives Models 564 23.4 The Assumptions 565 23.5 The Single-Period Model 570 Arbitrage-Free Evolutions 570 Zero-Coupon Bond Prices and Martingales 573 Understanding the Equal Pseudo-probability Condition 575 Actual versus Pseudo-probabilities 576 Caplet Pricing 576 The Hedge Ratio (the Holy Grail) 581 Risk-Neutral Valuation 581 Valuing a Floorlet 582 Valuing Various Interest Rate Derivatives 584 Multiple Factors 584 23.6 Summary 585 23.7 Appendix 586 The Equal Pseudo-probability Condition 586 Proof of Sufficiency 587 Proof of Caplet and Floorlet Parity 588

CONTENTS

23.8 Cases 589 23.9 Questions and Problems 589

CHAPTER 24 Multiperiod Binomial HJM Model

................. 592

24.1 Introduction 593 24.2 The Assumptions 593 24.3 The Two-Period Model 597 Arbitrage-Free Evolutions 597 Zero-Coupon Bond Prices and Martingales 599 Caplet Pricing 601 Floorlet Pricing 605 Valuing Various Interest Rate Derivatives 606 Multiple Factors 606 24.4 The Multiperiod Model 607 24.5 Forwards, Futures, and Swaptions 607 Forward Rate Agreements 607 Eurodollar Futures 608 Comparing Forward and Futures Rates 610 Swaptions 611 24.6 Summary 612 24.7 Appendix 613 Linking the Binomial HJM Model and the HJM Libor Model 613 24.8 Cases 615 24.9 Questions and Problems 616

CHAPTER 25 The Heath–Jarrow–Morton Libor Model

.... 618

25.1 Introduction 619 25.2 Why Caplets? 619 25.3 A History of Caplet Pricing 520 Black’s Model 620 The Heath–Jarrow–Morton Model 621 The HJM Libor Model 621 EXTENSION 25.1: Black’s Model for Pricing Commodity Options and Caplets 621 25.4 The Assumptions 622

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25.5 Pricing Caplets 627 Caplet and Floorlet Pricing (CFP) Data 627 Caplet Pricing Model 627 25.6 The Inputs 631 Observable Inputs 631 The Average Forward Rate Volatility 632 25.7 Understanding the HJM Libor Model 637 Forward Rates and Martingales 637 Risk-Neutral Valuation 637 Actual versus Pseudo-probabilities 638 25.8 Caps and Floors 638 Caps 639 Floorlets 639 Caplet–Floorlet Parity 639 Floors 641 25.9 Using the HJM Libor Model 641 The Greeks 642 Delta Hedging 643 Gamma Hedging 646 25.10 American Options, Futures, Swaptions, and Other Derivatives 647 25.11 Summary 647 25.12 Appendix 649 25.13 Cases 652 25.14 Questions and Problems 652

CHAPTER 26 Risk Management Models

................................ 655

26.1 Introduction 656 26.2 A Framework for Financial Risk Management 658 26.3 Computing the Loss Distribution 660 26.4 Value-at-Risk and Scenario Analysis 662 Value-at-Risk 662 Scenario Analysis 666 EXTENSION 26.1: Risk Measures 667 26.5 The Four Risks 668 Market Risk 668 Credit Risk 669 Structural Models 670

CONTENTS

EXTENSION 26.2: Real Options 672 Reduced-Form Models 673 EXTENSION 26.3: Credit Default Swaps 676 Liquidity Risk 677 Operational Risk 678 26.6 The Future of Models and Traded Derivatives 679 Model Risk 679 Derivatives 680 26.7 Summary 680 26.8 Cases 681 26.9 Questions and Problems 681

APPE N DI X A Mathematics and Statistics A.1 Exponents and Logarithms 685 Exponents 685 Logarithms 685 Continuous Compounding 687 A.2 The Binomial Theorem 687 A.3 The Normal Distribution 688 A.4 A Taylor Series Expansion 691 Single Variables 691 Multiple Variables 692 Glossary 693 References 704 Notation 710 Additional Sources and Websites 712 Books on Derivatives and Risk Management 715 Name-Index 718 Subject-Index 720

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Preface to Second Edition

We are pleased by the global reception of the first edition. The book has seen classroom usage in over 100 courses in 75 universities in 22 countries. It has been used widely as: (a) an introductory textbook for undergraduates (in business schools, in Arts & Sciences, as well as other programs), (b) an introductory text for MBAs, MS, and other masters students, (c) a recommended text in derivatives courses, and (d) a reference text in specialized courses.

Changes in the Second Edition Based on feedback from colleagues, students, and our own teaching experience we focused on updating institutional facts, reducing the historical and institutional material, and cleaning typos. A Solutions Manual for end-of-chapter Questions and Problems remains available for lecturers, together with PowerPoint slides and a Test Bank. These ancillaries can be downloaded from https://www.worldscientific.com/worldscibooks/ 10.1142/y0018-sm Although our free-software PRICED! still works in most computers, we find that many professors and students are more comfortable developing their own pricing models through Excel (we also make our Excel files available to adopters). Consequently, we continue to make PRICED! available but we do not provide any support. Lecturers and students who wish to download PRICED! can visit https://www.worldscientific.com/worldscibooks/10.1142/y0018 and click on the “Supplementary” tab.

Acknowledgements We owe many thanks to colleagues and students for their help, support, and suggestions. We especially thank two colleagues for their extraordinary help and support: Scott Fung (California State University, East Bay) and Thijs van der Heijden (University of Melbourne). We also thank Binay Bhushan Chakrabarti, Prem Chandrani, Ronald Ehrenberg, Craig Holden, Sreenivas Kamma, Ravindra Patil, and Zhenyu Wang for their comments, help, and support. We thank Yu Yan who translated our book into Chinese.

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PREFACE TO SECOND EDITION

We thank the following colleagues who, our records show, have adopted the book for classroom use as a required or recommended reading: Sofiane Aboura, Université de Paris XIII, Sorbonne Paris Cité; Vedat Akgiray, Bogaziçi ̆ Üniversitesi; Ali Akyol, University of Melbourne; Joao Freire de Andrade, Universidade NOVA de Lisboa; Kostas Andriosopoulos, ESCP Europe, London; Kevin Aretz, University of Manchester; C Bülent Aybar, Southern New Hampshire University; Giovanni Barone-Adesi, Università della Svizzera italiana at Lugano; Basabi Bhattacharya, Jadavpur University; Nalinaksha Bhattacharyya, University of Alaska Anchorage; Jedrzej Bialkowski, University of Canterbury; Ben Chongyang Chen, University of Memphis; Hsin-Fu Chen, Lunghwa University of Science & Technology; William Cheung, Waseda University; Paul Moon Sub Choi, Ewha Womans University; Steven P Clark, University of North Carolina at Charlotte; Francisco José Climent Diranzo, Universitat de València; Vladislav Damjanovic, University of Essex; Theodoros Diasakos, University of Stirling; Sina Erdal, University of Essex; Shih-Chuan Fang, Chaoyang University of Technology; Scott Fung, California State University East Bay; Gianluca Fusai, City, University London; Neal Galpin, University of Melbourne; George P Gao, T Rowe Price; Rajesh Ghai, University of Texas at Dallas; Chinmoy Ghosh, University of Connecticut; Bruce Grundy, University of Melbourne; Anthony Hall, University of Technology, Sydney; Joongho Han, Sungkyunkwan University; Barry Harrison, Nottingham Trent University; Mia Hinnerich, Stockholms Universitet; Darien Huang, Cornell University; Nagaratnam Jeyasreedharan, University of Tasmania; Timothy C Johnson, Heriot-Watt University; Jeffrey Jolley, California State University Fullerton; Andreas Kaeck, University of Sussex; Hui-Chuan Kao, Ling Tung University; Cenk Cevat Karahan, Bogaziçi ̆ Üniversitesi; Roger Klee, Cleveland State University; Ioannis Kyriakou, City, University London; Pascal Letourneau, University of Wisconsin-Whitewater; Hung-Chi Li, National Cheng Kung University;

PREFACE TO SECOND EDITION

Hai Lin, Victoria University of Wellington; Ravi Mateti, Concordia University; João Amaro de Matos, Universidade NOVA de Lisboa; Hongghi Min, Korea Advanced Institute of Science and Technology; Nikos Nomikos, City, University London; Kurtay Ogunc, Texas A & M University-Commerce; Jacques Olivier, HEC Paris; Late Philip Palm, who was at University of Washington Bothell; Ging-Ginq Pan, National Pingtung University of Science and Technology; Christophe Perignon, HEC Paris; Ser-Huang Poon, University of Manchester; Marcel Prokopczuk, Zeppelin University; Paulo J M Rodrigues, Universiteit Maastricht; Juan Carlos Rodriguez, Tilburg University; Mehmet Saglam, ̆ University of Cincinnati; Rafael de Santiago, Universidad de Navarra; Ravi Sastry, Southern Methodist University; Norman Seeger, Vrije Universiteit Amsterdam; Elena Sernova, HEC Lausanne; Kaisheng Song, University of Texas at Dallas; Erling Steigum, BI Norwegian Business School; Liya Shen, University of Essex; Manouchehr Tavakoli, University of St Andrews; Siegfried Trautmann, Johannes Gutenberg Universitat Mainz; Jiao Tong, Robert Gordon University; Hui-Huang Tsai, National United University; Jen-Tsung Tsu, Chien Hsin University of Science and Technology; Calum Turvey, Cornell University; Eleuterio Vallelado Gonzalez, Universidad de Valladolid; Thijs van der Heijden, University of Melbourne; Dmitry Vedenov, Texas A & M University; Lee Wakeman, Ohio University; Nathan Walcott, Southern Methodist University; Costas Xiouros, BI Norwegian Business School; Lulu Zeng, Purdue University; J Feng Zhao, University of Texas at Dallas; Lin Zou, Texas Woman’s University. We thank our editor Jack Repcheck of W. W. Norton & Company and his colleagues Dorothy Cook, Edward Crutchley, Theresia Kowara, Lindsey Osteen, Lauren Quantrill, Eric Svendsen, Janise Turso, Dina Vakser, and others who have faciliated the global acceptance that the book enjoys today. After Jack’s sudden and sad death, we moved to World Scientific Publishing. Our special thanks to our new editor Yubing Zhai and her colleagues Shreya Gopi, Karimah Samsudin and others who have made possible a smooth transition and a fine production of the second edition.

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Preface to First Edition

“History has many cunning passages, contrived corridors,” wrote T. S. Eliot in the poem “Gerontion.” The history of options and futures can be so described. Derivatives have traded for centuries in small, over-the-counter markets in London, New York, and several cities of continental Europe. In 1688, a sophisticated rice futures market was established in Osaka, Japan, that thrived for 250 years. Modern commodity futures markets began in 1848 with the founding of the Chicago Board of Trade to help protect farmers from commodity price swings. After many placid decades, the calm was again broken with the inauguration of the world’s first financial futures market in 1972 at the Chicago Mercantile Exchange and in 1973 with the opening of the Chicago Board Options Exchange. Serendipitously, 1973 also saw the publication of the Nobel Prize–winning Black–Scholes–Merton (BSM) option pricing model, which further spurred market expansion by enabling better pricing and hedging. Sometimes viewed as harmful and sometimes viewed as beneficial by the financial press and the public, derivatives have nonetheless played a significant role in financial markets across the centuries. Following the 2007 credit crisis, derivatives are most recently playing the role of the harmful security again, and new financial regulations have been proposed to rein them in. Although perhaps well intentioned, these attacks on derivatives are hurled by those unfamiliar with their proper use. Derivative securities, if used properly, reduce risk and facilitate real economic growth in the economy. In today’s complex world, modern financial institutions cannot succeed without the use of derivatives for managing the varied risks of their assets and liabilities. An understanding of their proper usage comes through a careful study of derivatives, which brings us to the purpose for writing this book. This book was written to be the first book read on derivatives and not the last. Our aim has been to design a book that is closely connected to real markets, examines the uses of derivatives but warns against their abuses, and presents only the necessary quantitative material in an easily digestible form—and no more! Given our purpose, this book differs from all existing derivatives textbooks in several important ways: ■

First, it is an introduction. We wanted to create a textbook accessible to MBAs and undergraduates both in terms of the concepts and mathematics. Option pricing is normally thought of as a complex, mathematical, and difficult topic. Our experience is that this topic can be presented simply and intuitively.

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Second, it is about financial markets. We wanted to write an economics, not a quantitative, book on derivatives. Since the first textbooks on option pricing,1 derivatives textbooks have taken a quantitative approach to the topic and are often encyclopedic in presentation. Little if any effort was spent on the underlying economics. In contrast, an economics perspective relates the market structure to the assumptions underlying the models. An understanding of when to use and when not to use a model based on its assumptions must be included. Our book does this.

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Third, it is a book about risk management. Generally speaking, there are four risks to be managed: (1) market risk, which includes commodity (including equity) price risk, interest rate risk, and foreign currency price risk; (2) credit risk; (3) liquidity risk; and (4) operational risk. We walk you through these different risks, with an emphasis on market risk, and discuss how they can be managed in business as well as in one’s personal life.

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Fourth, not only do governments regulate the markets, but many government entities use derivatives to promote the public’s welfare. The book often discusses the relevant issues from a regulator’s point of view and from a public policy perspective.

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Fifth, another unique feature of this book is an intuitive and accessible presentation of the Heath–Jarrow–Morton (HJM) model, which is the most advanced as well as a widely used derivatives pricing model. To make this model accessible, we first present the classical option pricing theory centering on the BSM model in a userfriendly fashion. Our presentation of the BSM model, however, is done with an eye toward the HJM model, emphasizing those aspects of the BSM model that are needed later. Then, when we study the HJM model, less development is needed. This approach enables us to present the HJM model in a parallel fashion to that of the BSM model, so if you see one, you see them both!

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Sixth, option pricing has a fascinating history, filled with colorful people and events. We share this history with the reader. This history is obtained from century-old books (now, uniquely available via the Internet), recent books, newspaper and magazine articles, websites, and personal experiences.

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Seventh, we have included enrichment material for the advanced reader through the use of inserts and appendixes. Many of these inserts include current research insights not available in existing textbooks.

The organization of this book is purposely designed to facilitate its use in many courses relating to options, futures, derivatives, risk management, investments, fixed income securities, and financial institutions. Three major courses that can be taught from this book are: (1) derivatives, (2) futures and commodities, and (3) fixed income securities and interest rate derivatives. In a sense, there are many books within this one cover.

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Jarrow and Rudd (1983) and Cox and Rubinstein (1984).

PREFACE TO FIRST EDITION

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Derivatives. This is a standard course on basic derivatives that gives an introduction to forwards, futures, options, and swaps but excludes interest rate derivatives for the most part. For this, use chapters 1–20 and most of Chapter 22. This would be for first-year MBAs and masters of financial engineering and for upper-level undergraduates in business schools, engineering schools, and economics departments.

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Futures and commodities. This is a course on futures and commodities, excluding option pricing. For this, use Chapters 1–6 and 8–13. The target audience is as for the prior item.

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Fixed-income securities and interest rate derivatives. This would be a models-based approach to teaching this material. Integrating ongoing research developments, Jarrow has been teaching such a course at Cornell University for over twenty years. For this, use Chapters 1, 2, 4, 6, 9, and 15–26. This course can be taught to MBAs and more mathematically inclined upper level undergraduates.

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Case-based courses. We have also recommended cases at the end of each chapter so that instructors can easily develop a case study–oriented derivatives course.

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Spreadsheet-based courses. Unlike other textbooks, we do not provide black boxes in which you input data and get a derivative price! In line with current teaching trends, we have woven spreadsheet applications throughout the text. Our aim is for students to achieve self-sufficiency so that they can generate all the models and graphs in this book via Excel. In addition, spreadsheet software—called Priced!—especially designed for this textbook is available to facilitate learning and to teach the course material.

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Use in courses in other areas. Courses in accounting and law can use this as background material.

This book is a “download” of our understandings of markets and derivatives obtained from decades of research, teaching, and consulting. The material has been class tested at the Cornell University, Helsinki School of Economics and Business Administration, Hong Kong University of Science and Technology, the Indian Institutes of Management at Ahmedabad and at Calcutta, Indiana University (Bloomington), University of Colorado at Boulder, and the University of North Carolina at Chapel Hill. We hope that the reader will have a better understanding of derivative securities and risk management models after reading this book. Although we have tried to make this textbook error free, please notify us if you discover any errors.

The Ancillaries This text is accompanied by a number of important ancillaries, each intended to enhance the learning experience for the student and the teaching experience for the instructor.

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For Students and Instructors PRICED! Developed by a Cornell computer science Ph.D. Tibor Jánosi specifically for this text and in collaboration with book authors Robert Jarrow and Arkadev Chatterjea, Priced! is Excel-based software that makes the models real rather than hypothetical. The software computes prices and hedge ratios for the four key derivative security models contained in the book: (1) the binomial model, (2) the Black–Scholes– Merton (BSM) model, (3) the discrete time Heath–Jarrow–Morton (HJM) model, and (4) the HJM Libor model. The prices and hedge ratios are represented visually—both graphically and in trees—for easy analysis by students and instructors. For quick recognition and clarity, the software’s inputs and outputs are color-coded. Priced! is also completely dynamic; when changes are made to inputs, outputs are updated instantaneously. This software can be used to illustrate all of the key concepts associated with the derivative models discussed in the course. Instructors can illustrate how a model’s prices and hedge ratios change when inputs are changed. Students can even use Priced! and current market prices from the financial press to compute actual prices.

For Students SOLUTIONS MANUAL Written entirely by the text’s authors, Robert Jarrow and Arkadev Chatterjea, the solutions manual provides completely worked solutions for all the problems included with the book. ISBN: 978-0-393-92094-9

For Instructors POWERPOINTS Created by coauthor Arkadev Chatterjea. The slides include lecture slides and all art from the book. There is also a separate set of PowerPoints created by Robert Jarrow for a fixed income course based on selected chapters in the book. Downloadable from wwnorton.com/instructors. TEST BANK Written by the text’s authors, Robert Jarrow and Arkadev Chatterjea. Downloadable formats available on wwnorton.com/instructors. ■

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Acknowledgments Bob thanks Peter Carr, Dilip Madan, Philip Protter, Siegfried Trautmann, Stuart Turnbull, and Don van Deventer for many conversations about derivatives over the years. For the many helpful discussions, comments, and material that helped improve this book, Bob and Arka would both like to thank Jeffery Abarbanell, Sobhesh K. Agarwalla, Jiyoun An, Warren B. Bailey, Gregory Besharov, Sommarat Chantarat,

PREFACE TO FIRST EDITION

Surjamukhi Chatterjea, Soikot Chatterjee, Sudheer Chava, Paul Moon Sub Choi, Steve Choi, Hazem Daouk, Werner Freystätter, Suman Ganguli, Nilanjan Ghosh, Michael A. Goldstein, Jason Harlow, Philip Ho, Michael F. Imhoff, Keon Hee Kim, Robert C. Klemkosky, Junghan Koo, Hao Li, Banikanta Mishra, Debi P. Mohapatra, Gillian Mulley, B. V. Phani, George Robinson, Ambar Sengupta, Asha Ram Sihag, Yusuke Tateno, and Han Zheng. We thank our family and friends for supporting us during this long project. In particular, Kaushik Basu, Nathaniel S. Behura, Amitava Bose, Alok Chakrabarti, Jennifer Conrad, Judson Devall, Ram Sewak Dubey, David Easley, Diego Garcia, Robert H. Jennings, Surendra Mansinghka, Robert T. Masson, Indranil Maulik, Tapan Mitra, Peter D. McClelland, Uri M. Possen, Erik Thorbecke, and Daniel J. Wszolek. We were fortunate to have a number of highly conscientious formal reviewers of the manuscript as it was taking shape. Their astute advice had a major impact on the final version of the manuscript. We cannot thank these reviewers enough: Farid AitSahlia (University of Florida), Gregory W. Brown (The University of North Carolina at Chapel Hill), Michael Ferguson (University of Cincinnati), Stephen Figlewski (New York University), Scott Fung (California State University, East Bay), Richard Rendleman (Dartmouth College), Nejat Seyhun (University of Michigan), Joel Vanden (The Pennsylvania State University), Kelly Welch (The University of Kansas), and Youchang Wu (University of Wisconsin– Madison). Finally, but certainly not least, special thanks to the editorial staff at Norton who have helped turn our manuscript into the finished book you are now reading: Hannah Bachman, Cassie del Pilar, Jack Repcheck, Nicole Sawa, and Amy Weintraub.

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I

Introduction

CHAPTER 1

Derivatives and Risk Management

CHAPTER 2

Interest Rates

CHAPTER 3

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Stocks

CHAPTER 6

Arbitrage and Trading

CHAPTER 7

Financial Engineering and Swaps

CHAPTER 4

Forwards and Futures

CHAPTER 5

Options

1 Derivatives and Risk Management 1.1 Introduction Copyright © 2018. World Scientific Publishing Company Pte. Limited. All rights reserved.

1.2 Financial Innovation

1.5 The Regulator’s Classification of Risk

Expanding Derivatives Markets

1.6 Portfolio Risk Management

Two Economic Motives

1.7 Corporate Financial Risk Management

1.3 Traded Derivative Securities EXTENSION 1.1 The Influence of Regulations, Taxes, and Transaction Costs on Financial Innovation Diverse Views on Derivatives Applications and Uses of Derivatives A Quest for Better Models

1.4 Defining, Measuring, and Managing Risk

Risks That Businesses Face Nonhedged Risks Risk Management in a Blue Chip Company

1.8 Risk Management Perspectives in This Book 1.9 Summary 1.10 Cases 1.11 Questions and Problems

INTRODUCTION

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1.1

Introduction

The bursting of the housing price bubble, the credit crisis of 2007, the resulting losses of hundreds of billions of dollars on credit derivatives, and the failure of prominent financial institutions have forever changed the way the world views derivatives. Today derivatives are of interest not only to Wall Street but also to Main Street. Derivatives are cursed as one of the causes of the Great Recession of 2007–2009, a period of decreased economic output and high unemployment. But what are derivatives? A derivative security or a derivative is a financial contract that derives its value from an underlying asset’s price, such as a stock or a commodity, or even from an underlying financial index like an interest rate. A derivative can both reduce risk, by providing insurance (which, in financial parlance, is referred to as hedging), and magnify risk, by speculating on future events. Derivatives provide unique and different ways of investing and managing wealth that ordinary securities do not. Derivatives have a long and checkered past. In the 1960s, only a handful of individuals studied derivatives. No academic books covered the topic, and no college or university courses were available. Derivatives markets were small, located mostly in the US and Western Europe. Derivative users included only a limited number of traders in futures markets and on Wall Street. The options market existed as trading between professional traders (called the over-the-counter [OTC] market) with little activity. In addition, cheating charges often gave the options market disrepute. Derivatives discussion did not add sparkle to cocktail conversations, nor did it generate the allegations and condemnations that it does today. Brash young derivatives traders who drive exotic cars and move millions of dollars with the touch of a computer key didn’t exist. Although Einstein had developed the theory of relativity and astronauts had landed on the moon, no one knew how to price an option. That’s because in the 1960s, nobody cared, and derivatives were unimportant. What a difference the following decades have made! Beginning in the early 1970s, derivatives have undergone explosive growth in the types of contracts traded and in their importance to the financial and real economy. According to the Bank for International Settlements (known as the BIS), the markets are now global and measured in trillions of dollars. Indeed, as depicted in Figure 1.1, in December 2016 the total outstanding US dollar notional value for exchange-traded derivatives was (26,172 + 41,072 =) 67,245 billion and for OTC derivatives a staggering 500,419 billion. Hundreds of academics study derivatives, and thousands of articles have been written on the topic of pricing derivatives. Colleges and universities now offer numerous derivatives courses using textbooks written on the subject. Derivatives experts are in great demand. In fact, Wall Street firms hire PhDs in mathematics, engineering, and the natural sciences to understand derivatives—these folks are admirably called “rocket scientists” (“quants” is another name). If you understand derivatives, then you know cool stuff; you are hot and possibly dangerous. Today understanding derivatives is an integral part of the knowledge needed in the risk management of financial institutions.

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CHAPTER 1: DERIVATIVES AND RISK MANAGEMENT

FIGURE 1.1: Global Derivatives Market Derivatives Notional Amounts (December 2016, in billions of US dollars) 10015 1350 6140

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27864

26172 41073 68598

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386356

Futures (Exchange-Traded)

Options (Exchange-Traded)

Foreign Exchange Contracts (Global OTC)

Interest Rate Contracts (Global OTC)

Equity-Linked Contracts (Global OTC)

Commodity Contracts (Global OTC)

Credit Derivatives (Global OTC)

Other Derivatives (Global OTC)

Unallocated (Global OTC) Source: https://www.bis.org/statistics/extderiv.htm, Table D1: Exchange-traded futures and options, by location of exchange; and https://www.bis.org/statistics/derstats.htm, Table D5: Global OTC derivatives market.

Markets have changed to accommodate derivatives trading in three related ways: the introduction of new contracts and new exchanges, the consolidation and linking of exchanges, and the introduction of computer technology. Sometimes these changes happened with astounding quickness. For example, when twelve European nations replaced their currencies with the euro in 2002, financial markets for eurodenominated interest rate derivatives sprang up almost overnight, and in some cases, they quickly overtook the dollar-denominated market for similar interest rate derivatives. This chapter tells the fascinating story of this expansion in derivatives trading and the controversy surrounding its growth. An understanding of the meaning of financial risk is essential in fully understanding this story. Hence a discussion of financial risk comes next, from the regulator’s, the portfolio manager’s, and the corporate financial manager’s points of view. We explain each of these unique perspectives,

FINANCIAL INNOVATION

using them throughout the book to increase our understanding of the uses and abuses of derivatives. A summary completes the chapter.

1.2

Financial Innovation

Derivatives are at the core of financial innovation, for better or for worse. They are the innovations to which columnist David Wessel’s Wall Street Journal article titled “A Source of Our Bubble Trouble,” dated January 17, 2008, alludes:

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Modern finance is, truly, as powerful and innovative as modern science. More people own homes—many of them still making their mortgage payments—because mortgages were turned into securities sold around the globe. More workers enjoy stable jobs because finance shields their employers from the ups and downs of commodity prices. More genius inventors see dreams realized because of venture capital. More consumers get better, cheaper insurance or fatter retirement checks because of Wall Street wizardry.

Expressed at a time when most of the world was in the Great Recession, this view is challenged by those who blame derivatives for the crisis. Indeed, this article goes on to say that “tens of billions of dollars of losses in new-fangled investments [in derivatives and other complex securities] at the largest US financial institutions—and the belated realization that some of those Ph.D. wielding, computer-enhanced geniuses were overconfident in the extreme—strongly suggests some of the brainpower drawn to Wall Street would have been more productively employed elsewhere in the economy.” But derivatives have been trading in various guises for over two thousand years. They have continued to trade because, when used properly, they enable market participants to reduce risk from their portfolios and to earn financial rewards from trading on special skills and information. Indeed, derivatives help to advance or postpone cash flows (borrowing and lending), to accumulate wealth (saving), to protect against unfavorable outcomes (insurance or hedging), to commit funds to earn a financial return (investment), and to accept high risks in the hope of big returns (speculation or gambling), which often goes along with magnifying the scale of one’s financial returns (leverage). Financial markets grow and real economic activity prospers because derivatives make financial markets more efficient. This is a theme to which we return repeatedly throughout the book.

Expanding Derivatives Markets Many factors have fueled the growth of derivatives markets. These include regulatory reforms, an increase in international commerce, population growth, political changes, the integration of the world’s economy, and revolutionary strides in information technology (IT). The interrelated financial markets are now more susceptible to global shocks and financial crises. The financial world has become a mad, bad, and dangerous place—financially speaking! More pronounced business cycles, default by sovereign nations, high-risk leveraged bets by hedge funds, imprudent investment in complicated securities by unsophisticated investors, and fraudulent actions by rogue traders have the potential to shake financial markets to their core. Financial

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6

CHAPTER 1: DERIVATIVES AND RISK MANAGEMENT

regulators exist to help prevent these catastrophes from happening. And if used properly, derivatives can also help to mitigate their effects on aggregate wealth. To help achieve economic stability, the central bank of the US, the Federal Reserve System (often referred to as the Federal Reserve or simply the Fed), historically used monetary policy tools to keep interest rates stable. In 1979, the Fed also began targeting money supply growth. Despite this oversight, oil shocks and other supply-side disturbances created double-digit inflation rates in the 1970s and 1980s, which in turn led to double-digit US interest rates that wiggled more than ever before. These highly volatile interest rates created a need for securities to help corporations hedge this risk. Interest rate derivatives arose. The Chicago Board of Trade (CBOT; now part of the CME Group) developed the first interest rate derivative contract, the Ginnie Mae futures, in 1975 and the highly popular Treasury bond futures in 1977. The foreign exchange market is one of the world’s largest financial markets, where billions of dollars change hands daily. From the mid-1940s until the early 1970s, the world economy operated under the Bretton Woods system of fixed exchange rates—all the currencies were pegged to the US dollar, and the dollar was pegged to gold at $35 per ounce. This stable monetary system worked well for decades. However, problems arose when gold prices soared. Countries converted their currencies into dollars and bought cheap gold from the US at the bargain price of $35 per ounce, making huge profits. This was an arbitrage, a trade that makes riskless profits with no investment. Consequently, US gold reserves suffered a terrible decline. Because all currencies were tied to the dollar, the United States could not adjust the dollar’s exchange rate to fix the problem. Instead, US president Richard Nixon abandoned the Bretton Woods system in 1971. Currencies now float vis-à-vis one another in a so-called free market, although their values are frequently managed by central banks. Floating exchange rates are more volatile than fixed exchange rates, and to hedge this newly created currency risk, the huge foreign exchange derivatives market was created. In this regard, 1972 saw the opening of the International Monetary Market, a division of the Chicago Mercantile Exchange (CME or Merc; now part of the CME Group) to trade foreign currency futures. The world’s first exchange-traded financial derivatives contract was born! Given these regulatory changes and well-functioning interest rate and foreign currency derivatives markets, in the recent past, many economists believed that a new era of greater macroeconomic stability had dawned, dubbed the Great Moderation. During the two decades before the new millennium, fluctuations in the growth of real output and inflation had declined, stock market volatility was reduced, and business cycles were tamed. However, the tide soon turned. In 2007–2009, many nations were mired in the Great Recession, with declining economic output and large unemployment. Stock market volatility, as measured by the widely followed VIX Index, shot up from 10 percent to an astonishing 89.53 percent in October 2008.1 Volatility had returned with a vengeance!

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See the Chicago Board Options Exchange’s website, www.cboe.com/.

TRADED DERIVATIVE SECURITIES

Two Economic Motives Regulatory changes have powerful impacts on markets. In fact, economics Nobel laureate Merton Miller argued in a 1986 article that regulations and taxes cause financial innovation. The reason is because derivative securities are often created to circumvent government regulations that prohibit otherwise lucrative transactions. And because most countries tax income from different sources (and uses) at different rates, financial innovations are often designed to save tax dollars as well. The desire to lower transactions costs also influences financial innovation. This perspective comes from another Nobel Prize–winning University of Chicago economist, Ronald Coase. Financial institutions often devise derivatives so that brokerage commissions, the difference between a securities dealer’s buying and selling prices, are minimized (see Extension 1.1). This leads us to an (almost) axiomatic truth that will guide us throughout the book: trading moves to those markets where transaction costs and regulatory constraints are minimized.

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1.3

Traded Derivative Securities

A security, such as a stock or bond, gives its holder ownership rights over some assets and cash flows. It exists as a paper document or an electronic entry record and usually trades in an organized market. You may be familiar with bonds, which are debts of the issuer, and stocks, which give investors equity in the issuing company. Bonds and stocks require an initial investment. Most bonds pay back a promised amount (principal or par value) at maturity. Some bonds pay interest (coupons) on a regular basis, typically semiannually, while others are zero-coupon bonds that pay no interest but are sold at a discount from the principal. The investment in stock is never repaid. Stockholders usually get paid dividends on a quarterly basis as compensation for their stock ownership. Stock prices can increase and create capital gains for investors, and this profit is realized by selling the stock. Alternatively, stock prices can decrease and create capital losses. Stocks and bonds are often used to create new classes of securities called derivatives, and that’s where the variety comes in. As previously noted, a derivative security is a financial contract whose value is derived from one or more underlying assets or indexes—a stock, a bond, a commodity, a foreign currency, an index, an interest rate, or even another derivative security. Forwards, futures, and options are the basic types of derivatives. These are explained later in the book. Some common terminology will help us understand the various derivative contracts traded: 1. Real assets include land, buildings, machines, and commodities, whereas financial assets include stocks, bonds, and currencies—both real and financial assets have tangible values. 2. Notional variables include interest rates, inflation rates, and security indexes, which exist as notions rather than as tangible assets.

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EXTENSION 1.1: The Influence of Regulations, Taxes, and Transaction Costs on Financial Innovation In the old days, finance mainly consisted of legal issues, institutional description, and investment rules of thumb. This changed in the middle of the twentieth century, when financial economics sprang to life as an offshoot of economics. In a little over two decades, a new finance based on a rigorous analytics emerged. James Tobin and Harry Markowitz’s portfolio theory (late 1940s and early 1950s); Franco Modigliani and Merton Miller’s M&M propositions concerning the irrelevance of a firm’s capital structure and dividend policy (late 1950s and early 1960s); William Sharpe, John Lintner, and Jan Mossin’s capital asset pricing model (mid-1960s) and Fischer Black, Myron Scholes, and Robert Merton’s option pricing model (early 1970s) established the basic theories. All these works have been celebrated with the Sveriges Riksbank (Bank of Sweden) Prize in Economic Sciences in Memory of Alfred Nobel, popularly known as the Nobel Prize in Economics. Other Nobel laureates, including Kenneth Arrow, Ronald Coase, Gerard Debreu, John Hicks, Paul Samuelson, and a well known economist, John Maynard Keynes, also contributed to finance. Two of these Nobel laureates’ views concerned financial innovation.

Miller’s View: Regulations and Taxes Spur Financial Innovation

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What leads to revolutionary change in financial institutions and instruments? University of Chicago professor Merton Miller (1923–2000) argued that a major cause of financial innovation is regulations and taxes. Innovative financial securities are often created to avoid government regulations that prohibit otherwise lucrative transactions. And because most countries tax income from different sources (and uses) at different rates, derivative securities can be designed to save tax dollars. Miller (1986) gives several examples: ■

Regulation Q of the US placed a ceiling on the interest rate that commercial banks could pay on time deposits. Although this wasn’t a problem during much of the postwar period, US interest rates rose above this ceiling during the late 1960s and early 1970s. When this occurred, US banks started losing customers. US banks realized, however, that Regulation Q did not apply to dollar-denominated time deposits in their overseas branches, and they soon began offering attractive rates via Eurodollar accounts. Interestingly, Regulation Q has long been repealed, but the Eurodollar market still continues to flourish.

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In the late 1960s, the US government imposed a 30 percent withholding tax on interest payments to bonds sold in the US to overseas investors. Consequently, for non-US citizens, the market for dollar-denominated bonds moved overseas to London and other money centers on the continent. This created the Eurobond market that still continues to grow today.

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The British government restricted dollar financing by British firms and sterling financing by non-British firms. Swaps, transactions in which counterparties exchange one form of cash flow for another, were developed to circumvent these restrictions.

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It was found in 1981 that US tax laws allowed a linear approximation for computing the implicit interest for long-term deep discount zero-coupon bonds. This inflated the present value of the interest deductions so much that a taxable corporation could actually profit by issuing a zero-coupon bond and giving it away! Not surprisingly, US corporations started issuing zero-coupon bonds in large numbers. This supply dwindled after the US Treasury fixed this blunder.

TRADED DERIVATIVE SECURITIES

9

Coase’s View: Transaction Costs are a Key Determinant of Economic Activity

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In 1937, Ronald Coase, a twenty-seven-year-old faculty member at the London School of Economics, published a simple but profound article titled “The Nature of the Firm.” Coase argued that transactions incur costs, which come from “negotiations to be undertaken, contracts have to be drawn up, inspections have to be made, arrangements have to be made to settle disputes, and so on,” and firms often appear when they can lower these transaction costs. With respect to financial markets, this logic implies that market participants often trade where they can achieve their objectives at minimum cost. Financial derivatives are often created so that these costs are minimized as well. The lowering of transaction costs as an economic motivation was especially important during the 1990s and the new millennium. Changes in the economic and political landscape and the IT revolution have made it possible to significantly lower transaction costs, even eliminating age-old professions like brokers and dealers from many trading processes. For example, this motive was a major reason why traders migrated from Treasury securities to Eurodollar markets. Eurodollar markets, being free from Fed regulations and the peculiarities of the Treasury security auction cycle, have fewer market imperfections and lower liquidity costs.

Writing financial contracts on values or cash flows determined by future realizations of real asset prices, financial asset prices, or notional variables creates a derivative security. Early derivatives were created solely from financial and real assets. For example, in the 1960s, agricultural commodity–based futures were the most actively traded derivative contracts in the US. However, as the economy has evolved, notional variables and derivative securities based on these notional variables have also been created. Notional variables are often introduced to help summarize the state of the economy. Just open the Wall Street Journal, and you will be amazed by the variety of indexes out there. You will find not only regular stock price indexes, such as the Dow-Jones and Standard and Poor’s, but a whole range of other indexes, including those based on technology stocks, pharmaceuticals stocks, Mexican stocks, utility stocks, bonds, and interest rates. In addition, notional values are often useful for the creation of various derivatives. Perhaps the most famous example of this is a plain vanilla interest rate swap whose underlyings are floating and fixed interest rates.

Diverse Views on Derivatives It is easy to speculate with derivatives: buy them, take huge one-sided bets, and laugh on the way to the bank or cry on the way to bankruptcy! Imprudent risk taking using derivatives is not uncommon, as illustrated by huge losses at many institutions, including Procter and Gamble (P&G) and Barings Bank. People are uncomfortable with derivatives because they are complex instruments: difficult to understand, highly leveraged, and oftentimes not backed by sufficient collateral. High leverage means that small changes in the underlying security’s price can cause large swings in the derivative’s value.

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For these reasons, derivatives attract strong views from both sides of the aisle. The renowned investors Warren Buffett and Peter Lynch dislike derivatives. In his “Chairman’s Letter” in Berkshire Hathaway’s 2002 annual report, Buffett characterized derivatives as “time bombs, both for the parties that deal in them and the economic system.” Lynch (1989) once stated that options and futures on stocks should be outlawed. These concerns were vindicated by the hundreds of billions of dollars of derivatives-related losses suffered by financial institutions during 2007 and 2008, which contributed to the severe economic downturn. By contrast, former Fed chairman Alan Greenspan opined in a speech delivered before the Futures Industry Association in 1999 that derivatives “unbundle” risks by carefully measuring and allocating them “to those investors most able and willing to take it,” a phenomenon that has contributed to a more efficient allocation of capital. And in Merton Miller on Derivatives, the Nobel laureate (Miller 1997, ix) assessed the impact of the “derivatives revolution” in glowing terms:

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Contrary to the widely held perception, derivatives have made the world a safer place, not a more dangerous one. They have made it possible for firms and institutions to deal efficiently and cost effectively with risks and hazards that have plagued them for decades, if not for centuries. True, some firms and some financial institutions have managed to lose substantial sums on derivatives, but some firms and institutions will always find ways to lose money. Good judgment and good luck cannot be taken for granted.

Nobel laureate physicist David Gross provides a nice analogy for this disagreement. He views knowledge as expanding outward like a growing sphere and ignorance as the surface of that sphere. With respect to derivatives, we have accumulated significant knowledge over the past thirty years, but with respect to the causes of tsunami-like financial crises, there is still much for us to learn.

Applications and Uses of Derivatives Derivatives trade in zero net supply markets, where each buyer has a matching seller. When hedging with a derivative, the other side of the transaction (counterparty) may be using it for speculative reasons. Hedging and speculation are often two sides of the same coin, and it is a zero-sum game because one trader’s gain is the other’s loss. Example 1.1 illustrates these concepts.

EXAMPLE 1.1: Hedging and Speculation in a Derivative Transaction ■

April is the beginning of the corn-growing season, the commodity used in our example.

■

Consider Mr. Short, a farmer in the midwestern United States. Short combines land, labor, seeds, fertilizers, and pesticides to produce cheap corn. He hopes to sell his corn harvest in September.

■

Expertise in growing corn does not provide a crystal ball for forecasting September corn prices. If Short likes sleeping peacefully at night, he may decide to lock in the selling price for September corn when he plants it in April.

TRADED DERIVATIVE SECURITIES

■

To see how this is done, let’s assume that everyone expects corn to be worth $10.00 per bushel in September. Ms. Long, a trader and speculator, offers to buy Short’s corn in September for $9.95 per bushel, which is the forward price.

■

To remove his risk, Short readily agrees to this forward price. Together they have created a forward contract—a promise to trade at a fixed forward price in the future. Although Short expects to lose 5 cents, he is happy to fix the selling price. The forward contract has removed output price uncertainty from his business. Short sees 5 cents as the insurance premium he pays for avoiding unfavorable future outcomes. Having hedged his corn selling price, Short can focus on what he knows best, which is growing corn.

■

Ms. Long is also happy—not that she is wild about taking risks, but she is rational and willing to accept some unwanted risk, expecting to earn 5 cents as compensation for this activity. Later in the book, you will see how a speculator may manage her risk by entering into another transaction at a better price but on the other side of the market.

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This simple example illustrates a key principle of historical importance—it was precisely to help farmers hedge grain prices that the first modern derivatives exchange, the CBOT, was established in 1848. Even the bitterest critics of derivatives have acknowledged this beneficial role. Nowadays, the uses of derivatives are many. Consider the following possibilities: ■

Hedging output price risk: A gold-mining company can fix the selling price of gold by selling gold futures.

■

Hedging input cost: A sausage maker can hedge input prices by buying pork belly futures.

■

Hedging currency risk: An American manufacturer buying machines from Germany for which the payment is due in three months can remove price risk from the dollar–euro exchange rates by buying a currency forward contract.

■

Hedging interest rate risk: A pension fund manager who is worried that her bond portfolio will get clobbered by rising interest rates can use Eurodollar futures (or options on those futures) to protect her portfolio.

■

Protecting a portfolio against a market meltdown: A money manager whose portfolio has reaped huge gains can protect these gains by buying put options.

■

Avoiding market restrictions: A trader can avoid short selling restrictions that an exchange might impose by taking a sell position in the options or the futures market.

■

Ruining oneself: A wretch can gamble away his inheritance with derivatives.

Before you finish this book, you will have a better understanding of the role derivatives play in each of these scenarios, except for the last one. The proclivity to gamble and lose arises from the depths of the human psyche, of which we have no special understanding.

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A Quest for Better Models Great recognition for derivatives came in 1997, when Robert Merton and Myron Scholes won the Nobel Prize in Economics for developing the Black–Scholes– Merton (BSM) model for option valuation. A co-originator of the model, Fischer Black, died earlier and missed receiving the prize, which is not given posthumously. This formula has become a staple of modern option pricing. Although extremely useful—options traders have it programmed into their computers—the BSM model makes a number of restrictive assumptions. These restrictive assumptions limit the application of the model to particular derivatives, those for which interest rate risk is not relevant (the BSM assumptions are discussed in later chapters). Subsequent researchers have relaxed these restrictive assumptions and developed pricing models suitable for derivatives with interest rate risk. The standard model used on Wall Street for this application is the Heath–Jarrow–Morton (HJM) model. The HJM model can be used to price interest rate derivatives, long-lived financial contracts, and credit derivatives. Credit derivatives are among the newest class of derivatives to hit Wall Street. Started in the early 1990s, they now compose a huge market, and they are the subject of intense public debate following the 2007 credit crisis and their role in it. Why develop and study better pricing models? Suppose you trade a variety of securities many times a day. If your models are better, you have an edge over your competitors in determining fair value. Over many transactions, when trading based on fair value—selling above fair value and buying below—your edge will pay off. Investment banking and Wall Street firms have no choice but to use the best models. It’s just too important for their survival. Powerful models also help traders manage risk.

1.4

Defining, Measuring, and Managing Risk

Risk is an elusive concept and hard to define. In finance and business, risk can be defined and measured in many ways, none of them completely universal. Risk often depends on the user’s perspective. We begin by looking at risk through a regulator’s eyes. Next, we discuss risk from an individual’s as well as an institutional trader’s viewpoint. Finally, we discuss risk from a corporate management perspective.

1.5

The Regulator’s Classification of Risk

In 1994, the international banking and securities regulators issued guidelines for supervising the booming derivatives market. The Basel Committee on Banking Supervision and the International Organization of Securities Commissions (IOSCO) recommended that for a stable world financial system, the national regulators had to

THE REGULATOR’S CLASSIFICATION OF RISK

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ensure that banks and securities firms have adequate controls over the risks they incur when trading derivatives.2 The Basel Committee’s Risk Management Guidelines for Derivatives (July 1994, 10–17) identified the following risks in connection with an institution’s derivative activities. IOSCO’s Technical Committee also issued a similar paper at the time:3 ■

Credit risk (including settlement risk) is the risk that a counterparty will fail to perform on an obligation.

■

Market risk is the risk to an institution’s financial condition resulting from adverse movements in the level or volatility of market prices. This is the same as price risk.

■

Liquidity risk in derivative activities can be of two types: one related to specific products or markets and the other related to the general funding of the institution’s derivative activities. The former is the risk that an institution may not be able to, or cannot easily, unwind or offset a particular position at or near the previous market price because of inadequate market depth or disruptions in the marketplace. Funding liquidity risk is the risk that the institution will be unable to meet its payment obligations on settlement dates or in the event of margin calls (which, we explain later, is equivalent to coming up with more security deposits).

■

Operational risk (also known as operations risk) is the risk that deficiencies in information systems or internal controls will result in unexpected loss. This risk is associated with human error, system failures, and inadequate procedures and controls.

■

Legal risk is the risk that contracts are not legally enforceable or documented correctly.

Managing market or price risk is the subject of this book. Initially, this topic attracted the sole attention of academics and practitioners alike. It still remains the most important risk for us to understand and to manage. The other risks only appear when normal market activity ceases. Credit risk evaluation is currently a subject of advanced research. As we explain later, exchange-traded derivatives markets are so designed that they are nearly free from credit risk. Liquidity risk is a persistent problem for traders who choose markets in which securities are not easily bought and sold. Chapter 3 discusses what makes markets illiquid. Operational risk is a reality with which one has to live. It is part of running a business, and appropriate management checks and balances reduce it. Legal risk isn’t a problem for exchange-

2 In 1974, the Group of Ten countries’ central-bank governors established the Basel Committee on Banking Supervision. The Basel (or Basle) Committee formulates broad supervisory standards and recommends statements of best practice in the expectation that individual authorities will implement (see “History of the Basel Committee and Its Membership,” www.bis.org/bcbs/history.htm). The International Organization of Securities Commissions, which originated as an inter-American regional association in 1974, was restructured as IOSCO in 1983. It has evolved into a truly international cooperative body of securities regulators (see www.iosco.org/about/about). 3

For both reports, see www.bis.org/publ. These definitions are modified only slightly from those contained in the referenced report.

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traded contracts; however, it’s a genuine problem in OTC markets. We leave this topic for the courts and law schools. This jargon-laden Basel Committee report also cited the need for appropriate oversight of derivatives trading operations by boards of directors and senior management and the need for comprehensive internal control and audit procedures. It urged national regulators to ensure that firms and banks operate on a basis of prudent risk limits, sound measurement procedures and information systems, continuous risk monitoring, and frequent management reporting. The Basel Committee reports have been extremely influential in terms of their impact on derivatives regulation. We return to the issues again in the last chapter of the book, after mastering the basics of derivatives securities.

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1.6

Portfolio Risk Management

Let us examine some issues in connection with portfolio risk management from the perspective of an individual or institutional trader. One of the first pearls of wisdom learned in finance is that to earn higher expected returns, one has to accept higher risks. This maxim applies to the risk of a portfolio, called portfolio risk. This portfolio risk, when applied to a single security, can be broken into two parts: (1) nondiversifiable risk, which comes from market-wide sources, and (2) diversifiable risk, which is unique to the security and can be eliminated via diversification. Pioneering works by Markowitz (1952), Sharpe (1964), Lintner (1965), and Mossin (1966) gave us the capital asset pricing model—the alphas, the betas—ideas and concepts used by market professionals and finance academics. Modern portfolio theory encourages investors to construct a portfolio in a topdown fashion. This involves three steps: (1) do an asset allocation, which means deciding how to spread your investment across broad asset classes such as cash, bonds, stocks, and derivatives; (2) do security selection, which means deciding which securities to hold within each asset class; and (3) periodically revisiting these issues, rebalancing and hedging the portfolio with derivatives as appropriate. Portfolio risk management is critical for investment companies such as hedge funds and mutual funds. These financial intermediaries receive money from the public, invest them in various financial securities, and pass on the gains and losses to investors after deducting expenses and fees.

1.7

Corporate Financial Risk Management

Risks That Businesses Face To understand the risks that businesses face, let’s take a look at a typical company’s balance sheet, a vital accounting tool that gives a snapshot of its financial condition by summarizing its assets, liabilities, and ownership equity on a specific date. An asset provides economic benefits to the firm, whereas a liability is an obligation that requires payments at some future date. The difference between the two accrues

CORPORATE FINANCIAL RISK MANAGEMENT

TABLE 1.1: Risks That a Business Faces Assets Current assets ■ Cash and cash equivalents (interest rate risk)

■

Accounts payable (interest rate risk, currency risk)

Accounts receivable (interest rate risk, currency risk)

■

Financial liabilities (interest rate risk, currency risk)

Inventories (commodity price risk) Long-term assets ■ Financial assets (interest rate risk, market risk, currency risk)

■

■

■

■

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Liabilities

Pension fund obligations (interest rate risk, market risk) Equity ■ Ownership shares

Property, plant, and equipment (interest rate risk, commodity price risk)

to the owners of the company as shareholder’s equity (hence the identity assets equals liabilities plus shareholder’s equity). Depending on whether they are for the short term (one year or less) or for the long haul, assets and liabilities are further classified as current or noncurrent, respectively. Table 1.1 illustrates some risks that affect different parts of a firm’s balance sheet. We see that a typical company faces three kinds of risks: currency risk, interest rate risk, and commodity price risk. These are the three components of market risk: 1. If a big chunk of your business involves imports and exports or if you have overseas operations that send back profits, then exchange rate risk can help or hurt. This is a risk that you must understand and decide whether to hedge using currency derivatives. 2. No less important is interest rate risk. It is hard to find companies like Microsoft, with tens of billions of dollars in cash holdings that can be quickly deployed for value-enhancing investments. Most companies are cash strapped. Interest rate fluctuations therefore affect their cost of funds and influence their investment activities. Interest rate derivatives offer many choices for managing this risk. 3. Unless a financial company, most businesses are exposed to fluctuations in commodity prices. A rise in commodity prices raises the cost of buying inputs that may not always be passed on to customers. For example, if crude oil prices go up, so does the price of jet fuel and an airline’s fuel costs. Sometimes airlines levy a fuel surcharge, but it is unpopular and often rolled back. Another option is to hedge such risks by using oil-price derivatives.

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Nonhedged Risks Despite the development of many sophisticated derivatives useful for hedging (and speculation), some risks are difficult, if not impossible, to hedge. For example, it is very hard to hedge operational risk. Recall that operational risk is the risk of a loss owing to events such as human error, fraud, or faulty management. Although a bank can buy insurance to protect itself from losses due to fire, no insurance company will insure a bank against the risk that a trader presses the wrong computer button and enters the wrong bond trade. For examples of such operational risk losses, see Chapter 26. Other difficult or impossible to hedge risks include losses because of changes in commodity prices for which there is no futures contract trading. We will now take a bird’s-eye view of how a blue chip company uses derivatives for risk management.

Risk Management in a Blue Chip Company

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The SFAS (Statement of Financial Accounting Standards) No. 133, “Accounting for Derivative Instruments and Hedging Activities,” as amended, requires US companies to report all derivative instruments on the balance sheet at fair value and establishes criteria for designation and effectiveness of hedging relationships. Let’s take a look at Form 10-K (annual report) filed with the US Securities and Exchange Commission by Procter & Gamble (P&G), a giant company that owns some of the world’s best-known consumer product brands.4 P&G heavily uses derivatives for risk management. A careful reading of this annual report reveals several interesting characteristics of P&G’s risk management activity: ■

P&G is exposed to the three categories of risk that we just mentioned.

■

P&G consolidates these risks and tries to offset them naturally, which means some risks cancel each other. It then tries to hedge the rest with derivatives.

■

P&G does not hold derivatives for trading purposes.

■

P&G monitors derivatives positions using techniques such as market value, sensitivity analysis, and value at risk. When data are unavailable, P&G uses reasonable proxies for estimating volatility and correlations of market factors.

■

P&G uses interest rate swaps to hedge its underlying debt obligations and enters into certain currency interest rate swaps to hedge the company’s foreign net investments.

■

P&G manufactures and sells its products in many countries. It mainly uses forwards and options to reduce the risk that the company’s financial position will be adversely affected by short-term changes in exchange rates (corporate policy limits how much it can hedge).

■

P&G sometimes uses futures, options, and swaps to manage the price volatility of raw materials.

4

See P&G’s 2015 Annual Report (http://www.pginvestor.com/CustomPage/Index?KeyGenPage=1073748359)

RISK MANAGEMENT PERSPECTIVES IN THIS BOOK

■

P&G designates a security as a hedge of a specific underlying exposure and monitors its effectiveness in an ongoing manner.

■

P&G’s overall currency and interest rate exposures are such that the company is 95 percent confident that fluctuations in these variables (provided they don’t deviate from their historical behavior) would not materially affect its financial statements. P&G expects significant risk neither from commodity hedging activity nor from credit risk exposure.

■

P&G grants stock options and restricted stock awards to key managers and directors (employee stock options are valued by using a binomial model that you will learn to implement in Part III of this book).

How does one understand, formulate, and implement hedging and risk management strategies like the ones adopted by P&G? Read on, for that is a major purpose of this book.

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1.8

Risk Management Perspectives in This Book

The Basel Committee’s risk nomenclature has become the standard way that market participants see, classify, and understand different types of risks in derivatives markets. In this book, we restrict our focus mainly to market or price risk. For this we usually wear four separate hats and look at risk management from the perspectives of an individual trader, a financial institution, a nonfinancial corporation, and a dealer. In early chapters, we start by examining financial securities and their markets from an individual trader’s standpoint. This simple and intuitive approach makes it easier to understand the material. It’s also historically correct because many markets, including those for stocks, futures, and options, were started by individual traders. But today’s derivatives markets have become playing fields for financial institutions. This is no surprise because as an economy develops, a greater share of its gross domestic product comes from services, of which financial institutions are a major constituent. These institutions generally engage in sophisticated ways of investing. We discuss derivatives and risk management from a financial institution’s viewpoint because you may eventually be working for one, or at the very least, you are likely to have financial dealings with such institutions on a regular basis. The term institution is a catchall phrase that includes commercial and investment banks, insurance companies, pension funds, foundations, and finance companies such as mutual funds and hedge funds. In later chapters, we study swaps and interest rate derivatives, whose markets are the near-exclusive domain of financial (and nonfinancial) institutions. Nonfinancial companies engage in more real activity than financial companies. Nonfinancials give us food, develop medicines, build homes, manufactures cars, refine crude oil to create gasoline, generate electricity, provide air travel, create household chemicals, and make computers to save and express our ideas. They buy one or more inputs to produce one or more outputs, and different kinds of risk (including exchange rate risk, interest rate risk, and commodity price risk) can

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seriously affect their businesses. This takes us to financial engineering, which applies engineering tools to develop financial contracts to meet the needs of an enterprise. This is our third hat: looking at derivatives from a nonfinancial company’s risk management perspective. Sometimes, we take a dealer’s perspective. A dealer is a financial intermediary who posts prices at which she can buy (wholesale or bid price) or sell (retail or ask price) securities to her customers. Trying to make a living from the spread, or the difference between these two prices, the dealer focuses on managing books, which means carefully controlling her inventory of securities to minimize risk. Both an individual trader and a financial institution can play a dealer’s role in financial markets. These four perspectives aren’t ironclad, and we flit from one to another as the discussion demands. In the final analysis, this shifting from one category to another isn’t bad for the aspiring derivatives expert. In the words of the immortal bard William Shakespeare, from Hamlet (if we take the liberty of forgetting about spirits and instead apply this to the mundane), “there are more things in heaven and earth, Horatio, than are dreamt of in your philosophy.” So stay awake and study derivatives, develop a sense for risky situations, understand the markets, learn pricing models, and know their limitations.

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1.9

Summary

1. Derivatives are financial securities that derive their value from some underlying asset price or index. Derivatives are often introduced to hedge risks caused by increasingly volatile asset prices. For example, during the last three decades of the twentieth century, we have moved away from a regime of fixed exchange rates to a world of floating exchange rates. The market-determined foreign exchange rates increased their volatility, creating the need for foreign currency derivatives. 2. In today’s interconnected global economy, risks coming from many different sources can make or break businesses. Derivatives can both magnify risk (leverage and gambling) or reduce risk (hedging). Although gambling with derivatives remains popular in some circles, and derivatives mishaps grab newspaper headlines, most traders prudently use derivatives to remove unwanted risks affecting their businesses. 3. There are many types of risk. Besides market risk, the regulators are concerned about credit risk, liquidity risk, operational risk, and legal risk. Though unglamorous, these risks can eat away profits and jolt the running of smooth-functioning derivatives markets. 4. Derivatives are very useful for managing the risk of a portfolio, which is a collection of securities. Portfolio risk management is important for both individual investors and financial companies such as hedge funds and mutual funds. Many businesses are exposed to exchange rate risk, interest rate risk, and commodity price risk and use derivatives to hedge them. US companies report derivatives usage and exposure on Form 10-K filed annually with the Securities and Exchange Commission. For example, P&G reports that it is exposed to exchange rate risk,

QUESTIONS AND PROBLEMS

interest rate risk, commodity price risk, and credit risk. P&G consolidates risks and attempts to offset them naturally and tries to hedge the remaining risk with derivatives. 5. We look at risk management from several different perspectives: that of an individual trader, a financial institution, a nonfinancial corporation, and a dealer. These perspectives aren’t ironclad, and we flit from one to another as the discussion demands.

1.10

Cases

Hamilton Financial Investments: A Franchise Built on Trust (Harvard Business

School Case 198089-PDF-ENG). The case discusses various risks faced by a finance company that manages mutual funds and provides discount brokerage services. Grosvenor Group Ltd. (Harvard Business School Case 207064-PDF-ENG). The

case considers whether a global real estate investment firm should enter into a property derivative transaction to alter its asset allocation and manage its business. Societe Generally (A and B): The Jerome Kerviel Affair (Harvard Business

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School Cases 110029 and 110030-PDF-ENG). The case illustrates the importance of internal control systems in a business environment that involves a high degree of risk and complexity in the context of a derivatives trader indulging in massive directional trades that went undetected for over a year.

1.11

Questions and Problems

1.1. What is a derivative security? Give an example of a derivative and explain why

it is a derivative. 1.2. List some major applications of derivatives. 1.3. Evaluate the following statement: “Hedging and speculation go hand in hand

in the derivatives market.” 1.4. What risks does a business face? 1.5. Explain why financial futures have replaced agricultural futures as the most

actively traded contracts. 1.6. Explain why derivatives are zero-sum games. 1.7. Explain why all risks cannot be hedged. Give an example of a risk that cannot

be hedged. 1.8. What is a notional variable, and how does it differ from an asset’s price? 1.9. Explain how derivatives give traders high leverage. 1.10. Explain the essence of Merton Miller’s argument explaining what spurs

financial innovation.

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1.11. Explain the essence of Ronald Coase’s argument explaining what spurs

financial innovation. 1.12. Does more volatility in a market lead to more use of financial derivatives?

Explain your answer. 1.13. When the international banking regulators defined risk in their 1994 report,

what definition of risk did they have in mind? How does this compare with the definition of risk from modern portfolio theory? 1.14. What’s the difference between real and financial assets? 1.15. Explain the differences between market risk, credit risk, liquidity risk, and

operational risk. 1.16. Briefly present Warren Buffett’s and Alan Greenspan’s views on derivatives. 1.17. Consider the situation in sunny Southern California in 2005, where house

prices have skyrocketed over the last few years and are at an all-time high. Nathan, a software engineer, buys a second home for $1.5 million. Five years back, he bought his first home in the same region for $350,000 and financed it with a thirty-year mortgage. He has paid off $150,000 of the first loan. His first home is currently worth $900,000. Nathan plans to rent out his first home and move into the second. Is Nathan speculating or hedging? 1.18. During the early years of the new millennium, many economists described the

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past few decades as the period of the Great Moderation. For example, ■

an empirical study by economists Olivier Blanchard and John Simon found that “the variability of quarterly growth in real output (as measured by its standard deviation) had declined by half since the mid-1980s, while the variability of quarterly inflation had declined by about two thirds.”

■

an article titled “Upheavals Show End of Volatility Is Just a Myth” in the Wall Street Journal, dated March 19, 2008, observed that an important measure of stock market volatility, “the Chicago Board Options Exchange’s volatility index, had plunged about 75% since October 2002, the end of the latest bear market, through early 2007”; the article also noted that “in the past 25 years, the economy has spent only 16 months in recession, compared with more than 60 months for the previous quarter century.” a. What were the explanations given for the Great Moderation? b. Does the experience of the US economy during January 2007 to Decem-

ber 2010 still justify characterizing this as a period of Great Moderation? Report (1) quarterly values for changes in the gross domestic product, (2) quarterly values for changes in the inflation rate, and (3) the volatility VIX Index value during this period to support your answer. 1.19. Drawing on your experience, give examples of two risks that one can easily

hedge and two risks that one cannot hedge.

QUESTIONS AND PROBLEMS

1.20. Download Form 10-K filed by P&G from the company’s website or the

US Securities and Exchange Commission’s website. Answer the following questions based on a study of this report: a. What are the different kinds of risks to which P&G is exposed? b. How does P&G manage its risks? Identify and state the use of some

derivatives in this regard. c. Name some techniques that P&G employs for risk management. d. Does P&G grant employee stock options? If so, briefly discuss this program.

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What valuation model does the company use for valuing employee stock options?

21

2 Interest Rates 2.1 Introduction 2.2 Rate of Return

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2.3 Basic Interest Rates: Simple, Compound, and Continuously Compounded EXTENSION 2.1 Conventions and Rules for Rounding, Reporting Numbers, and Measuring Time

2.4 Discounting (PV) and Compounding (FV): Moving Money across Time EXTENSION 2.2 Moving Multiple Cash Flows across Time

2.5 US Treasury Securities 2.6 US Federal Debt Auction Markets

2.7 Different Ways of Investing in Treasury Securities The Treasury Auction and Its Associated Markets The Repo and the Reverse Repo Market EXTENSION 2.3 A Repurchase Agreement Interest Rate Derivatives

2.8 Treasury Bills, Notes, Bonds, and STRIPS 2.9 Libor versus a Libor Rate Index 2.10 Summary 2.11 Cases 2.12 Questions and Problems

INTRODUCTION

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2.1

Introduction

Gentlemen prefer bonds. Perhaps you are chuckling at our “misquotation” of the famous Marilyn Monroe movie title Gentlemen Prefer Blondes. In fact, former US treasury secretary Andrew Mellon made this observation about bonds many decades before the movie.1 In all likelihood, he said this because bonds are considered safer than stocks. Corporations may miss dividend payments on stocks, but bonds are legally bound to pay promised interest on fixed dates. If the interest payments are not paid, default occurs, and the corporations are vulnerable to lawsuits and bankruptcy. However, at this stage in our presentation, we will assume that all bonds considered have no default (credit) risk. This greatly simplifies the analysis. Consequently, the bonds considered in this chapter are best viewed as US Treasury securities. Credit risk is considered only later in the book (in Chapter 26), after we have mastered default-free securities. Most bonds make interest payments according to a fixed schedule. Hence bonds are also called fixed-income securities. Bonds are useful for moving money from one period to another. Does money retain its value over time? Not really. Inflation, the phenomenon of a rise in the general price level, chips away money’s buying power little by little, year by year. A popular measure of the US inflation rate is the change in the consumer price index (CPI). At each date, the CPI measures the price of a fixed bundle of nearly two hundred goods and services that a typical US resident consumes. Unless you are buying computers and electronic gizmos whose prices have drastically fallen because of technological advances, a hundred dollar bill doesn’t quite buy as much as it did ten years ago. To reduce the cost of inflation, money earns interest. Instead of stashing cash under a mattress, one can put it into a savings account and see it grow safely. What, then, is an interest rate? An interest rate is the rate of return earned on money borrowed or lent. Ponder and you realize that the cost of borrowing a hundred dollar bill is the interest of perhaps $5 per year that’s paid to the lender. Suppose that the inflation rate is 3 percent per year, which is close to the US inflation rate in the new millennium. If the interest rate for risk-free loans is 3.10 percent per year, then borrowing is incredibly cheap, while 30 percent for such loans would be awfully expensive. Interest rates partly compensate the lender for inflation and partly reward her for postponing consumption until a later date. There are many ways of computing interest rates. Simple, compound, and continuously compounded rates are the three basic kinds. Each interest rate concept has its own use in finance theory as well as in practice. There are interest rates for risky as well as riskless loans. The first interest rates studied in this book are risk-free rates. How do you find risk-free interest rates? United States Government Treasury securities (the “Treasuries”) are considered default-free because of the taxing authority of the mighty federal government. Risk-free interest rates of various maturities can be easily extracted from their prices. We briefly explain how to do this extraction. Then we explain how the US Treasury securities market works, and we describe the 1

Andrew William Mellon (1855–1937) was US secretary of the treasury under three presidents (1921– 32). Source: US Treasury’s website, www.ustreas.gov/education/history/secretaries/awmellon.html.

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bills, notes, bonds, STRIPS, and TIPS that trade in these markets. We conclude the chapter with a discussion of the London Interbank Offered Rate (libor) and a liborbased rate index, which is a key rate used in the global interest rate derivatives market. We begin by introducing the concept of a rate of return. This, in turn, will lead us to a discussion of the three basic types of interest rates.

2.2

Rate of Return

In finance, we constantly talk about rates of return computed from prices. Rates of return measure how much an investment has earned. For example, if you buy a security for $50 today and sell it for $52 after six months, then you have earned $2 on your investment. The rate of return earned on your investment over this time period is 52 − 50 = 0.04, or 4 percent (when multiplied by 100 to express as a percentage) 50 It is hard to compare rates of return unless we explicitly mention the time interval over which they are computed. We need to know whether we are computing these over six months, or thirty-seven weeks, or ninety-one days. Consequently, it is worthwhile to standardize by computing them over a year. In this case, the annualized rate of return is

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12 ( 6 ) × 0.04 = 0.08, or 8 percent per year A more precise value for the rate of return may be obtained by considering the actual number of days for which the money was invested using 365 days in a year (some computations use 366 days in case of a leap year). Assuming that there were 181 days in this six-month period, we find the (annualized) rate of return as 365 ( 181 ) × 0.04 = 0.0807, or 8.07 percent per year The capital gain, the difference between the selling (final) and buying (initial) price, is the $2 earned on this investment. But there could be other cash flows at intermediate dates. For example, bonds make coupon payments and stocks pay dividends. If our $50 investment also received $0.50 interest at some intermediate date, then the (annualized) rate of return is 52 + 0.50 − 50 365 ( 181 ) × ( ) = 0.1008, or 10.08 percent per year 50 A profit of $2.50 is made on this investment. Expenses like brokerage account fees can cut into your profit. A profit occurs when there’s an overall gain from an investment or business activity. Businesses define it as total revenue minus total costs and record a profit if this is a positive number and a loss if it’s negative. In finance, we define Profit or loss = (Selling price + Income) − (Buying price + Expenses)

(2.1)

BASIC INTEREST RATES: SIMPLE, COMPOUND, AND CONTINUOUSLY COMPOUNDED

where the selling price is the final price, the buying price is the initial price, and a profit or loss happens if the result is positive or negative, respectively.2 We generalize this to develop a formula for computing the annualized rate of return (which is also called arithmetic return).

RESULT 2.1 Annualized Rate of Return The annualized arithmetic rate of return is 365 Rate of return = ( T ) ×

Selling price + Income − Expenses − Buying price ) ( Buying price

=(

Profit or loss 365 × ) ( Buying price ) T

(2.2)

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where T is the time interval, measured by the number of days over which the investment is held, and Income and Expenses denote positive and negative cash flows, respectively, from the investment (Rate of return is often multiplied by 100 and expressed as a percentage).

We need some reasonable conventions regarding rounding and reporting numbers and for measuring time, which we use throughout the book. They are discussed in Extension 2.1.

2.3

Basic Interest Rates: Simple, Compound, and Continuously Compounded

An interest rate is the rate of return promised (in case the money is yet to be lent) or realized (in case the money has already been lent) on loans. There seem to be a zillion ways of computing interest. Financial institutions like banks and credit card companies have devised clever ways of charging interest that appear less than they really are. That’s why many governments came up with a standard yardstick to help customers understand interest costs. For example, in the United States, all lenders have to disclose the rate they are charging on loans in terms of an Annual Percentage Rate (APR), a rate that annualizes using a simple interest rate. 2

An exception is short selling (see Chapter 3), in which selling occurs first and buying second, but the expression retains the positive and negative signs associated with these expressions.

25

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CHAPTER 2: INTEREST RATES

EXTENSION 2.1: Conventions and Rules for Rounding, Reporting Numbers, and Measuring Time To maintain consistency and to minimize unwanted errors, we follow some rules for rounding, reporting numbers, and calculations. We usually round to four places after the decimal point when reporting results. In all calculations, before reporting the numbers in the text, we retain 16 digits for accuracy. Any differences between reported results based on rounded numbers and calculated results (manipulations of the actual numbers) are due to these rounding errors: ■

If the number in the fifth place is more than 5, then add 1 to the fourth digit, e. g., 0.23456 becomes 0.2346.

■

If the number in the fifth place is less than 5, then keep it unchanged, e. g., 0.11223 becomes 0.1122.

■

If it’s exactly 5 and there are numbers after it, then add 1 in the fourth place, e. g., 0.123452 becomes 0.1235.

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In some contexts, we need to round to more places after the decimal point. The context will indicate when this is appropriate. Though we typically report numbers rounded to four places after the decimal point, the final dollar result is usually rounded to two places after the decimal. The dollar sign “$” is usually attached only in the final answer. We sometimes omit dollar signs from the prices when the context is understood. Different markets follow different time conventions. Treasury bill prices use the actual number of days to maturity under the assumption that there are 360 days in a year. Many banks pay interest compounded daily. Swaps use simple interest applied to a semiannual period, and many formulas use a continuously compounded rate. Most models measure time in years. If we start computing at time 0 and the security matures at time T, then it has a life of T years. If we start our clock at time t, then the life is (T – t) years. We prefer to use the first convention in our formulas. Time periods are converted by the following conventions: ■

If the time period is computed in days, then use the exact number of days and assume 365 days in the year (unless noted otherwise), e. g., 32 days will be 32/365 = 0.0877 year.

■

If the time period is computed in weeks, use fifty-two weeks in the year, e. g., seven weeks will be 7/52 = 0.1346 year.

■

If the time period is computed in months, use twelve months in the year, e. g., five months will be 5/12 = 0.4167 year. To summarize, we employ the following rules and conventions.

Rules of Rounding and Conventions of Reporting Numbers and Measuring Time ■

Reported numbers are rounded to four places after the decimal point. All calculations are performed using 16 digit numbers, not the rounded numbers. Any differences between reported results based on rounded numbers and calculated results (manipulations of the actual numbers) are due to these rounding errors.

■

The final dollar result is rounded to two places after the decimal with the dollar sign attached only in the final answer.

■

Time is measured in years. If we start computing from time 0 and the security matures at time T, then it has a life of T years; if the clock starts at time t, then the security has a life of (T – t) years.

BASIC INTEREST RATES: SIMPLE, COMPOUND, AND CONTINUOUSLY COMPOUNDED

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There are many different ways of computing interest rates. For a certain quoted rate, the realized interest can vary with (1) the method of compounding, (2) the frequency of compounding (yearly, monthly, daily, or continuously compounded), (3) the number of days in the year (52 weeks; 360 days, or 365, or 366 for a leap year), (4) the number of days of the loan (actual or in increments of months), and (5) other terms and conditions (collect half of the loan now, the other half later; keeping a compensating balance). Rather than bombard you with a list of interest rate conventions for different markets, we focus on the three basic methods: simple, compound, and continuously compounded interest rates. Understanding these concepts will lead to discounting and compounding, which will enable us to transfer cash flows across time. A simple interest rate is used in some sophisticated derivatives like caps and swaps. Money is not compounded under simple interest rates. An annual rate is quoted. When computed over several months, a fraction of the annual rate is used. Compound interest is when interest is earned on both the original principal and the accrued interest. Banks offer interest on daily balances kept in your account. When compounding interest, divide the annual rate by the number of compounding intervals (daily, weekly, monthly) and multiply these interest components together to compute the loan value at maturity. Continuous compounding pays interest on a continuous basis. One can view continuous compounding as the limit of compound interest when the number of compounding intervals gets very large and the time between earning interest gets very small! Example 2.1 demonstrates these three methods of interest computation.

EXAMPLE 2.1: Simple, Compound, and Continuously Compounded Interest Rates

Simple Interest Rates ■

Suppose you invest $100 today, and 6 percent is the simple interest rate per year. - A year from now, it will grow to $106. - After six months, it will grow to 100 × [1 + 0.06(1/2)] = $103. - After one day, it will be 100 × [1 + 0.06(1/365)] = 100.0164 = $100.02. - After 250 days, it will be 100 × [1 + 0.06(250/365)] = $104.11.

■

Summoning the power of algebra, we replace numbers with symbols to develop a formula for money growth under simple interest. Here L = $100 is the principal, which is the original amount invested. It is invested at the rate of interest i = 6 percent per year for T years, where T is measured in years or a fraction of a year. In the preceding expressions, T takes the values 1, 1/2, 1/365, and 250/365, respectively. Consequently, under simple interest, L dollars invested at i percent per year becomes after one year L(1 + i) and after T years L(1 + iT). This is illustrated in Figure 2.1. Notice that the simple interest rate is also the rate of return. This is verified by plugging the values in relation to Equation 2.2 to get (365/365)[L(1 + iT) − L]/L = i as T is one year.

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CHAPTER 2: INTEREST RATES

Compound Interest Rates ■

What about interest earning interest? This leads to the notion of compounding, which involves computing interest on both the principal and the accumulated interest. We split the time period T over which we are investing our money into m intervals. Within each interval, money earns simple interest. At the end of each interval, the principal and accumulated interest become the new principal on which interest is earned. If the interest is compounded semiannually, one year from now, $100 will grow to3 1 1 100 × [1 + 0.06 ( )] × [1 + 0.06 ( )] 2 2 1 2 = 100 × [1 + 0.06 ( )] 2 = $106.09 Compounding three times a year, one year from now, it will grow to 1 3 100 × [1 + 0.06 ( )] = $106.12 3 What about daily compounding that your local bank offers? With daily compounding, a year from now, the amount invested will grow to 100 × [1 + 0.06 (

1 1 × [1 + 0.06 ( … 365 times )] 365 365 )]

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= 100 × [1 + 0.06 (

365 1 365 )]

= $106.18 Now consider a loan for T years. Continuing with our example of daily compounding, after 250 days (or 250/365 year), $100 will grow to 100 × [1 + 0.06 (

1 1 × 1 + 0.06 ( … 250 times 365 )] [ 365 )]

= 100 × [1 + 0.06 (

250 1 365 )]

= $104.19 Rewriting the second line of the preceding expression with an eye toward generalization, we get 250

(365)( 1 365 ) 100 × [1 + 0.06 ( )] 365

(The book’s appendix gives the rules underlying these manipulations) ■

As before, we replace numbers with symbols to develop a formula for computing compound interest. Let L = $100 be the principal, which is invested at i = 6 percent per year for T years; however,

BASIC INTEREST RATES: SIMPLE, COMPOUND, AND CONTINUOUSLY COMPOUNDED

the interest is compounded m times every year. In the preceding expressions, m takes the values 2, 3, and 365, respectively. T = 1 year, except in the last example where it takes the value (250/365). Consequently, under compound interest, L dollars invested for one year becomes L(1 + i/m)m , and when invested for T years, it becomes (which is also shown in Figure 2.1) L (1 +

i mT m)

FIGURE 2.1: Simple, Compound, and Continuously Compounded Interest Rates

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Simple Interest (Interest does not earn interest) Time 0

1 year

L Investment

L(1 + i) Value after 1 year

Time T

L(1 + iT) Value at time T

Compound Interest (Interest earns interest) Time 0

Time 1/m

Time 2/m

1 year

L Investment

L(1 + i/m)

L(1 + i/m)2

L(1 + i/m)m Value after 1 year

Time T

L(1 + i/m)mT Value at time T

Continuously Compounded Interest (Interest earns interest continuously) Time 0

1 year

L Investment

Ler1 Value after 1 year

Time T

LerT Value at time T

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30

CHAPTER 2: INTEREST RATES

Continuously Compounded Interest Rates ■

But why stop at 365? Why not compound 1 million times a year? Then the investment grows to $106.1836545 at year’s end. Two things are happening here. As the compounding frequency increases, so does earned interest, but it grows at a decreasing rate—the money doesn’t grow very fast after a while. For example, compounding 10 million times will only give $106.1836547 at year’s end.

■

We can talk about “continuous compounding,” where the interest rate r (we now use r instead of i to denote the interest rate) is continuously compounded. This happens when the annual frequency of compounding m becomes larger and larger but the interest rate charged r/m becomes smaller and smaller during each of the compounding intervals. In that case, $1 becomes 1er at year’s end, where the exponential function e is the base of the natural logarithm (mathematically, we define log x ≡ ln (x) = M as equivalent to eM = x). It is approximated as 2.7183. Calculators have this as ex or exp ( … ), denoting the exponential function.

■

But that was for one year. The dollar investment grows to erT after T years. To summarize, under continuous compounding, L dollars invested at r percent per year for one year becomes Ler , and when invested for T years, it becomes LerT . (The appendix shows how this happens, and Figure 2.1 illustrates the result in a diagram.) In our example, this will give at year’s end 100e0.06 = $106.1836547

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Notice that the APR remains at 6 percent but the effective annual interest rate = er − 1 = 6.1837 percent. This is an example of a continuously compounded return (or logarithmic return). Continuously compounded interest rates are used as an input in the Black–Scholes–Merton model. Our example suggests that continuous compounding (which gives $106.18 at year’s end) is a much better approximation to daily compounding (which also gives $106.18) than is simple interest (which only gives $106). 3

The appendix discusses how to write expressions involving exponents (or indexes). It also gives rules for manipulating such expressions. Notice that the APR remains 6 percent, but your investment will earn more than this at year’s end. This is captured by the concept of an effective annual interest rate (EAR), which expresses the interest rate realized on a yearly basis. This is given by (1 + i/m)m - 1, where m is the frequency of compounding. Here, because m is 2, the EAR is 0.0609 or 6.09 percent.

We collect these three interest computation methods as Result 2.2.

DISCOUNTING (PV) AND COMPOUNDING (FV): MOVING MONEY ACROSS TIME

RESULT 2.2 Simple, Compound, and Continuously Compounded Interest Rates Simple Interest A principal of L dollars when invested for T years becomes L (1 + iT)

(2.3)

where i percent per year is the simple interest rate.

Compound Interest L dollars invested for T years becomes L[1 + (i/m)]mT

(2.4)

where i percent per year is the compound interest rate and m is the number of times the interest is compounded every year.

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Continuously Compounded Interest L dollars invested for T years becomes LerT

(2.5)

where r percent per year is the continuously compounded interest rate and e is the exponential function.

2.4

Discounting (PV) and Compounding (FV): Moving Money across Time

We can use interest rates to find the dollar return or to price a zero-coupon bond. If we invest a dollar today and it continues to earn interest that we do not withdraw, then the final amount on the maturity date is the dollar return. The dollar return measures how an invested dollar grows over time. In real life, you can earn a dollar return by investing in a money market account (mma), which earns the risk-free rate in each period.

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CHAPTER 2: INTEREST RATES

The inverse of a dollar return is the price of a zero-coupon bond. A zero-coupon bond (or zero) sells at a discount, makes no interest (or coupon) payments over the bond’s life, and pays back the principal (or the par value) on the maturity date. A zero-coupon bond and a mma are closely related securities; the price of a zerocoupon bond (B) and the dollar return (1 + R) obtained from investing in a mma are inverses of one another, B ≡ 1/(1 + R). Both mmas and zeros help us move funds across time. Example 2.2 shows how to compute a dollar return and the price of a zero.

EXAMPLE 2.2: Computing a Dollar Return, Pricing a Zero, and Moving Funds across Time ■

Suppose that the simple interest rate is 6 percent per year. Today is time 0, and the bond matures at time T = 1/2 year. Then the dollar return after six months of investing in a mma is 1 + R = 1 × [1 + 0.06 (1/2)] = $1.03 This dollar return is the future value of $1 invested today.

■

Today’s price of a zero-coupon bond that pays $1 after six months is B ≡ 1/ (1 + R) = 1/1.03 = $0.9709 = $0.97

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This is the present value of $1 received after six months.

RESULT 2.3 Compounding and Discounting Cash Flows with Dollar Returns and Zero-Coupon Bonds Consider cash flows C(0) available today (time 0) and C(T) available at some later date (time T). Then multiplication of today’s C(0) by the dollar return (or equivalently, division by a zero-coupon bond price) gives the future value C (0) (1 + R) = C (0) /B

(2.6)

Multiplication of the future C(T) by a zero-coupon bond price (or division by the dollar return) gives the present value (or discounted value) BC (T) = C (T) / (1 + R)

(2.7)

DISCOUNTING (PV) AND COMPOUNDING (FV): MOVING MONEY ACROSS TIME

where (1 + R) is the dollar return at time T from investing $1 today, B ≡ 1/(1 + R) is today’s price of a zero-coupon bond that pays $1 at time T, and T is the time to maturity in years. See Figure 2.2. In the case of simple interest, 1 + R = (1 + i × T)

(2.8)

where i is the simple interest rate per year. In the case of continuously compounded interest, 1 + R = erT

(2.9)

where r is the continuously compounded interest rate per year.

FIGURE 2.2: Compounding and Discounting Cash Flows using the Dollar Return (1 + R) or the Zero-Coupon Bond Price (B)

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Now (Time 0)

Future (Time T)

C(0)

C(0)(1 + R) = C(0)/B

BC(T) = C(T)/(1 + R)

C(T)

You can easily extend this result to move multiple cash flows across time, which may occur at different time periods (see Extension 2.2). Example 2.3 demonstrates

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CHAPTER 2: INTEREST RATES

how to compound and discount when a continuously compounded risk-free interest rate (r) is given.

EXAMPLE 2.3: Computing Present Values Using a Continuously Compounded Rate ■

Suppose that we receive $100 after six months. Let the continuously compounded risk-free interest rate r be 6 percent per year. Consequently, C(T) = $100, r = 0.06, and T = 0.5 year. Using expressions (2.4b) and (2.4d), the present value (PV) of $100 is 100 e−0.06 × 0.5 = 97.0446 = $97.04

■

Using the notation defined previously, B = $0.970446. If you invest this at a continuously compounded rate of 6 percent, you will get $1 in 6 months. Hence we can also write the PV of $100 in six months as 100B.

■

Notice that (1 + R) = 1/B = 1.0304545. Consequently, expressions (2.4a) and (2.4d) give the future value (FV) of $100 after six months as

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100 (1 + R) = $103.05

Continuous compounding will be useful in powerful options pricing models like the Black–Scholes–Merton model. If you are trying to compute options prices and you know the market price of a zero, then, as explained in Example 2.4, you can extract the risk-free continuously compounded rate and plug it into the Black– Scholes–Merton model.

EXTENSION 2.2: Moving Multiple Cash Flows across Time Moving cash flows across time is a fundamental tool that we use throughout the book. One can use interest rates, zero-coupon bonds, and mma values to do this.

EXT. 2.2 EX. 1: Moving Multiple Cash Flows across Time ■

Suppose you graduate a year from now and expect to get a job from which you squirrel away $5,000 at year’s end. Moreover, you expect a year-end bonus of $10,000. Assuming that we can commit “several sins”—(1) disregard the basic principle that risky cash flows must be discounted by risky discount rates and (2) operate under the assumption that the same interest rate applies to loans of different maturities—let us borrow against these future cash flows and use the funds to help pay your college tuition.

35

DISCOUNTING (PV) AND COMPOUNDING (FV): MOVING MONEY ACROSS TIME

■

Your bank offers you a loan at 6 percent interest, compounded daily. Approximating this by continuous compounding, we let r = 6 percent be the continuously compounded risk-free interest rate. Then the price of a zero-coupon bond maturing in t = 2 years is given by B(2) = 1/ert = e-rt = e-0.06 × 2 = $0.8869 (rounded to 4 decimal places). Writing C1 (2) = $5,000 and C2 (2) = $10,000, discounting the individual cash flows and adding them up gives B(2)C1 (2) = 0.8869 × 5,000 = $4,434.60 and B(2)C2 (2) = 0.8869 × 10,000 = $8,869.20, whose sum is B (2) C1 (2) + B (2) C2 (2) = $13, 303.81 (2.10)

■

Alternatively, you can add up the cash flows and then discount by multiplying by the zero-coupon bond price: B (2) [C1 (2) + C2 (2)] = 0.8869 × (5, 000 + 10, 000) = $13, 303.81

(2.11)

Both approaches give the same answer. In the actual calculations, B(1) and B(2) are not rounded to 4 decimal places (see Extension 2.1). ■

The equality of expressions (1) and (2) can be generalized to yield a formula for discounting cash flows belonging to a particular time period, t = 2: B (2) C1 (2) + B (2) C2 (2) = B (2) [C1 (2) + C2 (2)]

(2.12)

You can express this compactly by using the summation sign (∑). Then expression (3) can be written as 2

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B (2) ■

2

Cs (2) = B (2) Cs (2) [∑ ] ∑ s=1 s=1

(2.13)

What about extending this result to cash flows available at different time periods? Continuing with our previous example, suppose your grandparents have been saving for your college education by investing in a tax-favored education savings account. Your paternal grandparents will gift you C1 (1) = $12,000 and your mother’s parents will give you C2 (1) = $10,000 at time t = 1. Assuming that the interest rate r is 6 percent, and the price of a zero-coupon bond maturing in one year is B(1) = e-0.06 × 1 = $0.9418, the present value of these cash flows is B (1) [C1 (1) + C2 (1)] = 0.9418 × (12, 000 + 10, 000) = $20, 718.82

■

Suppose you want to determine how much of a loan you can take out today based on these future cash flows. For this you need to compute the present value of the four cash flows, two after one year and two after two years. They have the same value, $34,022.63, irrespective of how you add them up: B (1) C1 (1) + B (1) C2 (1) + B (2) C1 (2) + B (2) C2 (2) = B (1) [C1 (1) + C2 (1)] + B (2) [C1 (2) + C2 (2)]

■

A generalization yields our next result.

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CHAPTER 2: INTEREST RATES

RESULT 2.4 The Sum of a Present Value of Cash Flows Is Equal to the Present Value of the Sum Suppose that there are S securities, which provide s cash flows (s = 1, 2, … , S ), some of which could be zero. These cash flows can occur at times t (t = 1, 2, … ,T), so that the cash flow from security s at time t is Cs (t). Then the sum of the present value of each cash flow is equivalent to (1) adding up the cash flows from all securities at a particular time t, (2) computing the PV of this sum, and (3) adding up these cash flows across time. This can be expressed as T

S

∑∑ t=1 s=1

T

B (t) Cs (t) =

∑ t=1

S

B (t)

[∑ s=1

Cs (t) ]

(2.14)

where B(t) is the price of a zero-coupon bond that pays $1 at time t.

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EXAMPLE 2.4: Finding the Continuously Compounded Rate from a Zero-Coupon Bond ■

Suppose a zero-coupon bond that is worth $0.90 today pays $1 after two years. Assuming that money grows at the continuously compounded rate of r percent per year, we have 0.90e2r = 1 or, e2r = 1/0.90

■

Taking natural logarithms and remembering that log(eX ) = X, 2r = log (1/0.90) or r = (1/2) log (1/0.90) = 0.0527 or 5.27 percent

Replacing numbers with symbols in the preceding expression (2 = T and 0.90 = B = 1/[1 + R]) gives us Result 2.4, which allows us to move from a notional variable (a posted interest rate) to a traded asset price (a zero-coupon bond or a dollar return), and vice versa.

US TREASURY SECURITIES

RESULT 2.5 Computing Continuously Compounded Interest Rates from a Zero-Coupon Bond Price or a Dollar Return The continuously compounded interest rate per year r can be computed from a zero-coupon bond price B (or a dollar return 1 + R) by r = (1/T) log (1/B) = (1/T) log (1 + R)

(2.15)

where T is the time to maturity in years.

Now that you have seen how to price zero-coupon bonds (B), move funds across time (T), find dollar returns (1 + R), and compute continuously compounded interest (r), you may wonder where zero-coupon bond prices come from. This brings us to the Treasury securities market, where debts of the US government trade.

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2.5

US Treasury Securities

Modern finance has made Treasury securities (Treasuries) interesting and exciting by developing a whole range of derivatives that depend on them. Although seemingly ordinary, Treasuries are the bedrock on which the world of finance is built. Following are five reasons why: 1. Five characteristics make Treasuries particularly important: (1) these debt securities have no default risk as they are backed by the full faith and credit of the US government; (2) they trade in a market with some of the smallest bid/ask spreads in the world, which is the transaction cost of buying and selling Treasuries (only 1 or 2 basis points for the most active issues, a basis point being 1/100th of 1 percent); (3) their interest payments are free from state and local taxes; (4) they have low minimum denominations starting at $100; and (5) they offer a spectrum of maturities that range from one day to thirty years. 2. The prices of Treasuries can help us determine borrowing and lending rates for future dates. What is the forward rate for borrowing money for two years starting 10 years from today? As Chapter 21 shows, the answer can be easily determined from Treasury prices. 3. We will study options and futures that are written on Treasuries. Remember the T-bond futures that we mentioned in the first chapter? They are one of the most actively traded futures contracts. The world has become a more volatile place, and

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CHAPTER 2: INTEREST RATES

derivatives based on fixed-income securities (bonds) can be very useful for hedging interest rate risks. 4. Interest rate options pricing models like the Heath–Jarrow–Morton model relax the constant interest rate assumption of the Black–Scholes–Merton model. They often use Treasury rates of different maturities as the necessary inputs. 5. Moreover, in part of a growing trend, financial firms convert many individual loans and debts into a package of securities using a process called securitization and sell them to third-party investors. Your mortgage loans, car loans, and credit card loans may have been financed that way. Loan access to wider markets generates more competitive rates that benefit consumers. The interest rates on these assetbacked securities are determined by adding a basis point spread to comparable maturity Treasury rates.

2.6

US Federal Debt Auction Markets

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When former US president Ronald Reagan was asked for his thoughts on the growing federal deficit, he replied, “The deficit is big enough to take care of itself!” No matter which party held power in Washington, D. C., the last several decades have seen government spending exceed taxes. Years of budget deficits (= Government receipts – Government expenditures) have led to a mountain of federal debt, standing at over $13 trillion at the end of the millennium (Figure 2.3 shows the level of public debt on an annual basis). This is hardly surprising. There is a natural tendency toward overspending in a democracy because elected politicians bring pork-barrel projects to help their constituencies. Whatever the reason, the US national debt and deficit became alarmingly large in the 1990s and took center stage on the political arena. US citizens interpreted continuing deficits as implying higher future taxes. Supply-side tax cuts of Reagan’s era lost their appeal, and federal income tax rates were raised immediately. A robust economy also increased tax receipts. Consequently, the US government started paying off its debts at the turn of the new millennium. But the surpluses soon evaporated, and the budget deficit is back in the red. As circumstances shape destiny, a huge federal debt has forced the Treasury to devise efficient ways of raising funds. Consequently, the United States has developed an extremely sophisticated system of selling Treasuries through sealed bid auctions, a model that has been adopted by many other nations. The buyers in such auctions submit sealed bids. These are promises to buy a fixed number of Treasuries at a fixed price. The Treasury awards the securities to the highest bidders. In reality, these are yield auctions, in which the bidders specify a yield and the Treasury translates this interest rate into a price. Every year, the Treasury finances the public debt through over 250 auctions, each typically selling $10 to $20 billion worth of securities.

US FEDERAL DEBT AUCTION MARKETS

FIGURE 2.3: U. S. Debt Outstanding 20,000 US Federal 18,000 Debt (in billions 16,000 of dollars) 14,000 12,000 10,000 8,000 6,000 4,000 2,000

97 19 99 20 01 20 03 20 05 20 07 20 09 20 11 20 13 20 15

95

19

93

19

91

19

89

19

87

19

85

19

83

19

81

19

79

19

19

19

77

0

Time

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Source: http://www.treasurydirect.gov/govt/reports/pd/histdebt/histdebt.htm.

Primary dealers are large securities firms with whom the New York Fed buys and sells Treasuries to conduct open-market operations that fine-tune the US money supply, examples include Goldman Sachs & Co. LLC, UBS Securities LLC, and Deutsche Bank Securities Inc. These firms also actively bid in Treasury auctions, as do other direct and indirect bidders. These bidders tread cautiously because bidding incorrectly, even several basis points, can make a difference between millions of dollars in profits and losses. So players in the auction submit bids as near to the bidding deadline as possible using all available information. Do all investors in Treasuries take part in competitive bidding? No, because the Treasury allows noncompetitive bids for smaller amounts. These bids are always filled, and they pay the price determined by the competitive bidders. Historically, 10 to 20 percent of a typical Treasury auction has been awarded to noncompetitive bidders. Noncompetitive bids encourage the direct participation of regular folks in the auction process. The quotes from competitive and noncompetitive bids are tallied, securities are allocated to the successful bidders, and the results of the auction are announced in a press conference. The Wall Street Journal and the other news providers carry the announcement.

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2.7

Different Ways of Investing in Treasury Securities

Bidding in auctions is not the only way to purchase Treasury securities. For example, you can buy Treasuries (1) in the when-issued and secondary markets, (2) through the repo and reverse repo markets, or (3) via interest rate derivatives. This section describes each of these alternative ways of buying Treasuries. Barring rare exceptions, these markets tend to be fairly liquid and are usually dominated by big institutional investors (institutions such as commercial banks, investment banks, pension funds, mutual funds, and hedge funds).

The Treasury Auction and Its Associated Markets

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A week or so before an auction, the Treasury announces the size of the offering, the maturities, and the denominations of the auctioned Treasuries. The Treasury permits forward trading of Treasury securities between the announcement and the auction, and the to-be-auctioned issue trades when, as, and if issued. Traders take positions in this when-issued market, and a consensus price emerges. The when-issued market spreads the demand over seven to ten days, which leads to a smooth absorption of the securities by the market. The buyer and seller in the when-issued market fulfill their commitments after the Treasuries become available through the auction. Newly auctioned Treasuries are called on-the-run and have lower spreads than off-the-run securities issued in prior auctions. A very active resale (secondary) market exists for Treasuries, giving investors further chances to invest. The timeline for the three markets for buying Treasuries is shown in Figure 2.4.

FIGURE 2.4: Timeline of Treasury Auction and Associated Markets

Treasury announcement

Auction

When-issued market

Settlement

Post-auction secondary market

DIFFERENT WAYS OF INVESTING IN TREASURY SECURITIES

41

The Repo and the Reverse Repo Market Investment banks operate a security dealership business that makes profits by trading standard securities as well as custom-made products for clients. They do this by highly leveraging both the asset and liability sides of their balance sheets (see Chapter 1 for a typical firm’s balance sheet). For example, believing that buying Treasuries will gain in value and borrowing Treasuries will decline in value, the firm may support $100 billion worth of purchased securities and $95 billion worth of borrowed securities using only $5 billion of capital. Leverages like this are usually accomplished through repo and reverse repo markets, examples of which are given in Extension 2.3. Partially contributing to the immense popularity of repos was the fact that they initially provided the only method of short selling Treasuries and corporate debt securities (see Extension 2.3). This short selling technique became antiquated when interest rate derivatives arrived. Derivatives facilitate the same transaction but far more efficiently. We will explore how this works in the final part of the book. Although this particular economic role of repos is now less important, the repo market still thrives and is an important element of Treasury markets.

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EXTENSION 2.3: A Repurchase Agreement Suppose that Repobank has purchased a large quantity of Treasury securities and it needs an overnight loan to finance the purchase. Repobank expects to sell the Treasuries on the market the next day, so an overnight loan is sufficient. To finance its position, it enters into a repurchase agreement (“does a repo”) with RevRepobank (another fictitious name), who enters into a reverse repo (“does a reverse”) transaction. The next example outlines the mechanics of this transaction in simple terms.

EXT. 2.3 EX. 1: Example of a Repo and a Reverse Repo ■

As per standard industry norms, Repobank takes, say, $10 million from RevRepobank and sells RevRepobank Treasury securities worth a little more. The next day, Repobank repurchases those securities at a slightly higher price—the extra amount determines an annual interest rate known as the repo rate. Though it involves a purchase and a repurchase, a repo is basically a short-term loan that is backed by high-quality collateral (see Ext. 2.3 Fig. 1).

■

If Repobank defaults, then RevRepobank keeps the securities. If RevRepobank fails to deliver the securities instead, then Repobank keeps the cash longer. In that case, the repo is extended by a day, but the terms remain the same—Repobank pays the day after tomorrow the amount it was supposed to pay tomorrow, essentially keeping the funds for an extra day at zero interest. And this is repeated day after day until the cash and the securities are exchanged. You may find that holders of scarce securities are able to borrow funds in the repo market at close to a zero percent rate of interest.

■

Repos provide a way of short selling Treasuries and corporate debt securities. After acquiring the securities, RevRepobank can short sell them to a third party. Later, it can buy these securities in the market, at a lower price if the bet is successful, and return them to Repobank and close out the repo.

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■

Entering into a repo agreement is equivalent to borrowing cash and using the short maturity Treasury security as collateral. The repo rate is the effective borrowing rate for Repobank and the lending rate for RevRepobank.

■

Repos are usually for overnight borrowing and are known as overnight repos. If Repobank wants to borrow funds for another night, then the whole process must be repeated. Alternatively, term repos are set up for a fixed period that lasts longer than overnight, whereas open repos have no fixed maturity and may be terminated by a notice from one of the sides. The market is huge. Major players include heavyweights like the Federal Reserve Bank, state and local governments, commercial and investment banks, mutual funds, and large companies.

EXT. 2.3 FIG. 1: An Overnight Repo and a Reverse Repo Transaction Starting Date (Today)

$10 million Repobank

RevRepobank

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T-securities worth $10 million

Ending Date (Tomorrow)

$10(1 + interest) million Repobank

RevRepobank T-securities worth $10 million

TREASURY BILLS, NOTES, BONDS, AND STRIPS

Interest Rate Derivatives Derivatives whose payoffs depend on Treasury (or closely related) securities are called interest rate derivatives.4 There are derivatives like futures on Treasuries and options on futures on Treasuries that allow investors to place leveraged bets or set up hedges on interest rates. We will discuss interest rate derivatives and their pricing models in Part IV of the book.

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2.8

Treasury Bills, Notes, Bonds, and STRIPS

The Treasury sells three types of marketable debts: (1) coupon bonds that pay interest (coupons) every six months and a principal amount (par or face value) at maturity, (2) zero-coupon bonds that don’t pay interest but pay back the principal at maturity, and (3) floating rate notes that pay interest quarterly. The United States issues bonds with maturities of one year or less in the form of zero-coupon bonds and calls them Treasury bills (or T-bills). Coupon bonds have two names: those with original maturity of two to ten years are called Treasury notes (or T-notes), whereas those with original maturity of more than ten years up to a maximum of thirty years are called Treasury bonds (or T-bonds). These notes and bonds make coupon and principal payments that remain fixed over the security’s life. Treasury notes of two year maturity, which pay interest quarterly based on the 13 week T-bill rate also are issued by the Treasury. There is an exception. In 1997, the Treasury started selling a new class of securities called inflation-indexed bonds or TIPS (Treasury Inflation Protected Securities). These bonds guarantee a fixed real rate of return, which is the nominal rate of return in dollar terms minus the inflation rate as measured by the CPI over their life. This is accomplished by raising the principal of the bond each year by changes in the US CPI. Each year, the coupon payment is computed by multiplying the adjusted (and changing) principal by the real rate of return. The name “coupon” may sound strange to your ears. In days gone by, bonds had coupons attached to them. The bearer of the bond detached the coupon and sent it to the issuer, who mailed back interest payments. Coupons are rare these days. In fact, since 1983, all Treasury securities are kept in book entry form at the Federal Reserve Bank’s computers, and the owner is given a receipt of ownership and receives deposits directly from the Fed. But the term coupon survives. The Treasury does not issue zeros of long maturities. Still, a pure discount bond with a single payment at maturity appeals to many investors. Wall Street firms figured out a way of making money by selling what people demanded. Wall Street firms bought coupon-bearing Treasury securities, put them in a trust to make them safe, and issued claims against the principal and the different coupon payments. This was an arbitrage opportunity: the firms paid less for the original Treasury than what they

4

Closely related to Treasury securities are Eurodollar deposits, which are discussed later in this chapter.

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CHAPTER 2: INTEREST RATES

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collected by selling the artificially created zero-coupon bonds. In February 1985, the Treasury entered this activity by allowing Treasury securities to be STRIPped. How do STRIPS (Separate Trading of Registered Interests and Principal of Securities) work? Although the Treasury does not issue or sell STRIPS directly to investors, it allows financial institutions as well as brokers and dealers of government securities to use the commercial book-entry system to separate a Treasury note or a bond’s cash flows into strips and sell them as individual zero-coupon bonds. Claims to individual cash flows coming out of a Treasury security are synthetically created zerocoupon bonds of different maturities. Moreover, an investor can buy up the individual strips from the market and reconstruct the original T-bond or note. Figure 2.5 shows how thirty-one strips can be created from a newly issued fifteen-year T-bond. A Treasury security can be stripped at any time from its issue date until its maturity date. For tracking purposes, each cash flow due on a certain date is assigned a unique Committee on Uniform Security Identification Procedures (CUSIP) number that depends on its source—consequently, they are categorized as coupon interest (ci), note principal (np), and bond principal (bp). Even though they may come from different Treasury securities, all ci due on a certain date have the same generic CUSIP and may be combined to create a coupon paying note or a bond. By contrast, np or bp is unique for a particular security and hence is not interchangeable. The STRIPS program had been very successful: 1. STRIPS made Treasuries more attractive to investors, leading to greater demand, higher prices, lower yields, and cheaper financing of the national debt. For example, not many people would be interested in holding a thirty-year bond with sixty cash flows. But a newly issued thirty-year bond can be stripped into sixty semiannual coupon payments and a final principal payment. These cash flows can be sold to different investors needing zeros of different maturities. For example, grandparents of a newborn may gift a strip for future education costs. 2. STRIPS help to identify the term structure of interest rates, a graph that plots the interest rate on bonds against the time to maturity. Earlier in this chapter, we talked about the benefits of knowing the risk-free rates of different maturities, an important input to the pricing of interest rate derivatives studied in Part IV of the book. Once we get the price of a Treasury security, we know the cash flows that it generates. Particularly useful are T-bill and STRIPS prices for moving cash flows across time.

TREASURY BILLS, NOTES, BONDS, AND STRIPS

FIGURE 2.5: Creation of STRIPS from a 15-Year Treasury Bond Newly Issued 15-Year T-Bond

Payments Time (years)

C ½

C 1

C 1½

…

…

…

C C and L 14½ 15

C denotes coupons and L principal. Holder gets all the cash flows. STRIP Creation

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Payments Time (years)

C ½

C 1

C 1½

…

…

…

C C and L 14½ 15

C and L sold as separate zero-coupon bonds (STRIPs).

Prices of Treasuries can be found in financial newspapers like the Wall Street Journal and Investor’s Daily. The quotes are collected from the over-the-counter market and are for transactions of $1 million or more. Notes, bonds, and strips are quoted in 32nds, so a quote of 93:08 (for bid) and 93:09 (for ask) for a strip maturing in one year means that $93 and 8/32 is the price the dealer will buy and $93 and 9/32 is the price the dealer will sell. These quotes are based on a par value equal to $100. T-bill prices follow an entirely different convention. They are quoted in terms of a banker’s discount yield. Before introducing this notion, it may be useful to see how the rate of return and the bond-equivalent yield are determined. When the annualized rate of return (see Result 2.1) is applied to Treasury bills, we get its bond-equivalent yield. This is reported as the “ask yield” in the Wall Street Journal, which corresponds to the bond-equivalent yield earned when buying a T-bill from a

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dealer. The bond-equivalent yield for Treasury bill prices is Bond equivalent yield = (

365 (1 − B) T ) B

(2.16)

where B is the bill’s price expressed as a bond that pays $1 at maturity and T is the number of days from settlement to maturity. However, Treasury bills are quoted in terms of a banker’s discount yield, which is typically reported for both ask and bid prices. Ask is the price at which one can readily buy a security from a dealer, and bid is the price at which one can promptly sell. They are analogous to prices at which a car dealer will sell or buy a used car. The banker’s discount yield differs from the bond-equivalent yield in two ways: (1) the denominator has the face value instead of the price paid (note the 1 in the denominator) and (2) the interest rate is annualized on a 360-day basis. The banker’s discount yield is given by Banker’s discount yield = (

360 (1 − B) T ) 1

T or Bill price, B = 1 − (banker’s discount yield) [ ( 360 )]

(2.17) (2.18)

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where 1 is the face value (par value) of the bill, B is bill price, and T is the number of days from settlement (the date ownership is transferred, time 0) to maturity. Example 2.5 applies these formulas to data from the financial press.

EXAMPLE 2.5: T-Bill Price Computation ■

The financial press reports the following: - Days to maturity 24 - Bid 4.71 percent - Ask 4.67 percent - Change 0.01 - Ask yield 4.75 percent

■

Using expression (2.7b) and assuming that the bill’s principal is $1, the ask price for the bill is = 1 − 0.0467 × (24/360) = $0.996887 In other words, $0.996887 is the ask price of a zero-coupon bond that pays $1 in twenty-four days. Similarly, $0.996860 is the bid price of a zero that pays $1 in twenty-four days.

LIBOR VERSUS A LIBOR RATE INDEX

■

How much are we actually earning on our investment? This is given by the bond-equivalent yield (reported as ask yield because it is computed from the ask price). Using expression (2.6) Bond − equivalent yield (for ask yield) = (365/24) × [(1/0.996887) − 1] = 0.0475 This verifies the quote.

The inverse relationship between bond prices and interest rates leads to an inversion of the relation between the ask/offer and bid for interest rates. In most markets, the ask/offer is higher than the bid, but that’s not the case for the ask and the bid when we quote them as a banker’s discount yield. Using the numbers from Example 2.4, this situation is shown in Figure 2.6.

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FIGURE 2.6: Inversion of Ask and Bid When Quoted as Banker’s Discount Yield (Inverted–bid above ask) Bid (as bdy) 4.71 percent

(As in most markets–ask above bid) Ask price $0.996887

Ask (as bdy) 4.67 percent

Bid price $0.996860

The banker’s discount yield is important because it helps price a T-bill. Using the face value in the denominator or 360 days in the computation doesn’t have any economic significance. It’s just the way pricing conventions have developed in this market. Using 360 days for quoting interest rates and using 1 in the denominator (as in Equation 2.2) just make it easier to compute.

2.9

Libor versus a Libor Rate Index

Though US Treasury securities are fundamental to the international bond markets, there are other interest rates that are keenly followed and widely used. In particular, interest rates earned on dollar deposits in non-US banks are such an example. These rates are related to the London Interbank Offered Rate (libor, pronounced lie-bore), which is the interest rate used to determine cash payments on most swap contracts in the swap market. Swaps are discussed in Chapters 7 and 22. To understand how libor gets created, let us peek at the London interbank market. Owing to their business activities, banks and other financial institutions face deficit or surplus fund situations on a regular basis. Banks in the United States manage

47

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48

CHAPTER 2: INTEREST RATES

their cash by borrowing or lending in the interbank federal funds market. Major London banks handle those imbalances by borrowing or lending deposits of different maturities in the London interbank market. The most important of these deposits are Eurodollars, which, as you may recall from Chapter 1, are US dollars held outside the United States in a foreign bank or a subsidiary of a US bank. Owing to the dual benefit of being dollar deposits free from US jurisdiction, Eurodollars have gained huge popularity among a whole range of holders, including central banks, financial institutions, companies, and retail individual investors. Now a bank with surplus funds lends the funds to another bank for a fixed time period at the libor valid for that period. This is a Eurodollar rate (say, 5.02 percent per year for a three-month deposit) in case of Eurodollars. Alternatively, a bank pays the London Interbank Bid Rate (libid, pronounced lie-bid) for the privilege of borrowing money: three-month libid could be 5.00 percent for Eurodollars. These rates may change minute by minute, and they may vary from bank to bank, but competition ensures that they are almost nearly the same at any given point in time. The Intercontinental Exchange (ICE) collects libor quotes from its panel of major banks for deposit maturities ranging from overnight to a year and computes a Libor rate index that is widely reported. For Eurodollar deposits, the ICE collects libor quotes from numerous banks, truncates some of the largest and smallest values, and, averages the rest to compute a libor rate index. It does a similar exercise for deposits denominated in many other currencies, announcing a term structure of Libor rate indexes for each different currency. This libor rate index is the most popular global benchmark for short-term interest rates, and it enters into numerous derivative contracts. An index is used instead of a particular bank’s Eurodollar rate because an index is less prone to manipulation. As the libor rate index has some credit risk, its values are larger than a similar maturity Treasury security, the difference being known as the Treasury/Eurodollar (TED) spread. Instead of Treasuries, many large financial institutions use the libor rate index as a proxy for the risk-free interest rate in derivative pricing models. Like Treasury securities, Eurodollar interest computations assume 360 days in a year. The Eurodollar market is now measured in trillions of dollars. In later chapters, we will study the vast interest rate derivatives market that uses Eurodollars as the underlying commodity.

2.10

Summary

1. An interest rate is the rate of return earned on money borrowed or lent. The annualized rate of return on an investment is given by 365 T ) Selling price + Income − Expenses − Buying price × ( ) Buying price

Rate of return = (

=(

Profit or loss 365 × T ) ( Buying price )

SUMMARY

where T is the time interval, measured by the number of days over which the investment is held, and Income and Expenses denotes any positive or negative cash flows, respectively, from the investment during this period. 2. There are many ways of computing interest. For a certain quoted rate, the realized interest can vary with (1) the method of compounding, (2) the frequency (yearly, monthly, daily, or continuously compounded), (3) the number of days in the year (52 weeks; 360 days, or 365, or 366 for a leap year), and (4) the number of days of the loan (actual or in increments of months). 3. The three basic methods of computing interest are simple, compound, and continuous compounding. In simple interest, an annual rate is used for the loan duration. When computed over several months, a fraction of that annual rate is used. In case of compound interest, interest is computed on both the original principal as well as the accrued interest. Continuous compounding involves paying interest on the principal and accrued interest on a continuous basis.

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4. Interest rates help us find the price of a zero-coupon bond (zeros). These bonds sell at a discount and pay back the face value at maturity. They pay no interest otherwise. Multiply any dollar amount by a zero-coupon bond’s price to get its present value, or divide the amount by it to get its future value. 5. US Treasury securities, whose payments are backed by the taxing power of the US government, are considered default-free. They help us determine the risk-free interest rate. They are sold through a sealed-bid auction, in which all successful bidders pay the same price as the lowest successful bid. One purchases a Treasury security by (1) trading in the auction and its associated markets, (2) entering into a repurchase (repos are short-term debts collateralized by high-quality debt securities) or a reverse repo transaction, or (3) investing in interest rate derivatives. 6. Marketable Treasury securities (Treasuries) are classified as bills, notes, and bonds. Treasury bills are zero-coupon bonds with original maturity of one year or less. Coupon bonds have two names: those with original maturity of two to ten years are called Treasury notes (or T-notes), whereas those with original maturity of more than ten years up to a maximum of thirty years are called Treasury bonds (or T-bonds). 7. Financial institutions and brokers and dealers of government securities can create zero-coupon bonds by segregating cash flows of Treasury notes and bonds and sell them to the public. These artificially created zeros are called STRIPS. 8. In the bond markets, the annualized rate of return is called the bond-equivalent yield. In case of Treasury bills, Bond equivalent yield = (

365 (1 − B) T ) B

where B is the price of a T-bill that pays $1 at maturity and T is the number of days from settlement (the date ownership is transferred, time 0) to maturity. But

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Treasury bills are quoted in terms of a banker’s discount yield, which is given by Banker’s discount yield = (

360 (1 − B) T ) 1

9. Besides Treasuries, major interest rates in the global financial markets include libor and repo rates. Libor and libid are the respective rates for the banks to lend and borrow surplus funds in the London interbank market. The ICE (1) collects libor quotes from major banks for deposits denominated in many different currencies, (2) for each currency it collects quotes with maturities ranging from one day to one year, (3) computes a trimmed mean by dropping some of the highest and lowest rates, and (4) releases these rates to the market, creating a Libor rate index. 10. A libor rate index for Eurodollar deposit constructed by the ICE (which are dollar-denominated deposits held outside the US and free from US banking laws and regulations) has emerged as the most popular global benchmark for shortterm interest rates and enters into numerous derivatives contracts. Many large financial institutions are replacing Treasuries with bbalibor as a proxy for the risk-free interest rate. Many floating interest rates are set at a spread above the relevant Treasury security rate or the Libor rate index.

2.11

Cases

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Breaking the Buck (Harvard Business School Case 310135-PDF-ENG). The case

educates students about how money market funds work and the challenges faced in managing these funds during the financial crisis of 2008–9. Foreign Ownership of US treasury Securities (Darden School of Business Case

UV1366-PDF-ENG, Harvard Business Publishing). The case describes the workings of the US Treasury securities market and explores the implications of significant foreign holdings of US debt. Innovation at the Treasury: Treasury Inflation-Protection Securities (A)

(Harvard Business School Case 204112-HCB-ENG). The case explores the various risks as well as policy issues surrounding the introduction of a new financial security product.

2.12

Questions and Problems

2.1. The interest rate is 5 percent per year. Compute the six-month zero-coupon

bond price using a simple interest rate. 2.2. The interest rate is 5 percent per year. Compute the six-month zero-coupon

bond price using a compound interest rate with monthly compounding. 2.3. The interest rate is 5 percent per year. Compute the six-month zero-coupon

bond price using a continuously compounded interest rate.

QUESTIONS AND PROBLEMS

Time (in years)

Cash Flows (in dollars)

0 (today)

105

1

7

2

9

3

108

2.4. The interest rate is 5 percent per year. Compute the six-month zero-coupon

bond price using a banker’s discount yield (the zero-coupon bond is a US T-bill with 180 days to maturity). 2.5. What is a fixed-income security? The next three questions are based on the

following table, where the interest rate is 4 percent per year, compounded once a year. 2.6. Compute the present value of the preceding cash flows. 2.7. Compute the future value of the preceding cash flows after three years. 2.8. What would be the fair value of the preceding cash flows after two years? 2.9. If the price of a zero-coupon bond maturing in three years is $0.88, what is

the continuous compounded rate of return? 2.10. What are the roles of the primary dealers in the US Treasury market?

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2.11. What is the when-issued market with respect to US Treasuries? What role does

this market play in helping the US Treasury auction securities? 2.12. What is the difference between on-the-run and off-the-run Treasuries? 2.13. What is a repurchase agreement? Explain your answer with a diagram of the

transaction. 2.14. What is a Treasury STRIPS? What benefits do the trading of Treasury STRIPS

provide? 2.15. Explain how a libor rate index is computed by the ICE. 2.16. What is a Eurodollar deposit, and what is a TED spread? 2.17. What is the difference between Treasury bills, notes, and bonds? What are

TIPS, and how do they differ from Treasury bills, notes, and bonds? 2.18. You bought a stock for $40, received a dividend of $1, and sold it for $41 after

five months. What is your annualized arithmetic rate of return? 2.19. Suppose that you are planning to enroll in a master’s degree program two years

in the future. Its cost will be the equivalent of $160,000 to enroll. You expect to have the following funds: ■

From your current job, you can save $5,000 after one year and $7,000 after two years.

51

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52

CHAPTER 2: INTEREST RATES

■

You expect a year-end bonus of $10,000 after one year and $12,000 after two years.

■

Your grandparents have saved money for your education in a tax-favored savings account, which will give you $18,000 after one year.

■

Your parents offer you the choice of taking $50,000 at any time, but you will get that amount deducted from your inheritance. They are risk-averse investors and put money in ultrasafe government bonds that give 2 percent per year.The borrowing and the lending rate at the bank is 4 percent per year, daily compounded. Approximating this by continuous compounding, how much money will you need to borrow when you start your master’s degree education two years from today?

3 Stocks 3.1 Introduction

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3.2 Primary and Secondary Markets, Exchanges, and Over-the-Counter Markets 3.3 Brokers, Dealers, and Traders in Securities Markets 3.4 Automation of Trading 3.5 The Three-Step Process of Transacting ExchangeTraded Securities 3.6 Buying and Selling Stocks Trading at the New York Stock Exchange Over-the-Counter Trading

Alternative Trading Systems: Dark Pools and Electronic Communications Networks

3.7 Dollar Dividends and Dividend Yields 3.8 Short Selling Stocks 3.9 Margin—Security Deposits That Facilitate Trading EXTENSION 3.1 Margin and Stock Trading

3.10 Summary 3.11 Cases 3.12 Questions and Problems

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CHAPTER 3: STOCKS

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3.1

Introduction

Institutions that play similar economic roles can differ widely in appearance. A farmer’s market in a remote African village and a fancy shopping mall in a North American suburb are both markets. MBA degree–holding sellers of customized derivatives products and peddlers of Oriental rugs are both dealers. The man at the airport who helps you find a cheap hotel near an exotic beach and the realtor who arranges the purchase of a vacation home are both brokers. As these examples illustrate, seemingly diverse economic phenomena often have commonalities. This leads to this chapter’s theme—“unity in diversity”; here we try to understand the features common to different securities markets and their traders.1 We introduce primary and secondary markets, exchanges and over-the-counter markets, brokers and dealers, and the bid and ask prices that dealers post. We also discuss market microstructure, a subfield of finance that studies how market organization and traders’ incentives affect bid/ask spreads. We classify traders into different categories on the basis of their trading strategies. We describe how automation is transforming trading and the three-step process of execution, clearing, and settlement that is common when transacting exchange-traded securities. Next, we study the effect of dividends on stock prices and portfolios, paving the way for managing a portfolio that replicates a stock index and pricing derivatives written on it. Finally, we illustrate the process of short selling stock and margins. The process of trading securities, the players who trade, and the facilitators who make trading possible are also present in the bond markets discussed in Chapter 2. However, we discuss them with respect to stock markets first because these are the easiest markets to understand—trading of different derivatives can also be seen as extensions of this process.

3.2

Primary and Secondary Markets, Exchanges, and Over-the-Counter Markets

As a small company grows, it needs more and more cash to finance its operations. After initial financing from founders and friends, and perhaps private equity from venture capitalists, the company becomes ready to tap the capital markets. A privately held company sells shares to the public for the first time in an initial public offering (IPO), which is the primary market for stocks. To issue stocks, you hire an investment banker who manages the process. Through cumbersome paperwork, publishing information brochures, and getting approval from government agencies, the investment banker burns a lot of your future proceeds in the process. Soon after their birth in primary markets, stocks move on to secondary markets, where they change ownership through secondhand transactions. Exchanges and the over-the-counter (OTC) market are the two basic types of secondary markets. 1

The phrase “Unity in Diversity” is the official motto of the European Union (“United in Diversity”), Ghana, Indonesia, South Africa, and some other nations; it has also been used to describe India.

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PRIMARY AND SECONDARY MARKETS, EXCHANGES, AND OVER-THE-COUNTER MARKETS

Traditionally, the main characteristic of an exchange was a central physical location where buyers and sellers gathered to trade standardized securities under a set of rules and regulations, the New York Stock Exchange (NYSE) being a prime example. However, the information revolution of the 1990s changed that forever— most exchanges in the new millennium, especially those outside of the US, do not have floor trading; instead, they have a centralized computer network that does trade matching. By contrast, any trade away from an exchange is called an OTC transaction.2 The expression over-the-counter goes back to the early days of the US, when banks primarily acted as dealers for stocks and bonds, which were sold at counters in their offices. There are organized markets for OTC transactions, where commercial and investment banks, institutional investors, brokers, and dealers participate. Such OTC markets (also called interbank markets) have no central location. Telephone, telex, and computer networks connect geographically dispersed traders and make trades possible in OTC markets. Exchanges used to handle the bulk of stock trading in the US, but in recent times, organized OTC markets have taken on increased importance. Exchanges operate under the basic assumption that if all trades take place in a central location, this provides a level playing field that leads to fair price formation. US stock exchanges are regulated by the Securities and Exchange Commission (SEC), a federal government agency whose job is to protect investors, punish violators, and prevent fraud. Before the SEC’s formation in 1934, the stock exchanges were considered a hotbed of shady activity that routinely hurt the ordinary investor. Mark Twain famously said in Pudd’nhead Wilson’s tale (1894), “October. This is one of the peculiarly dangerous months to speculate in stocks. The others are July, January, September, April, November, May, March, June, December, August and February.” Interestingly, many folks no longer consider stock exchanges as bad or risky; rather, they are now viewed as venues where respectable investors pursue wealth by taking acceptable levels of risk. The derivatives exchanges, riding a wave of mysticism and intrigue, have easily snatched the disrepute. All exchange-traded securities, including derivatives, must satisfy government requirements before they can be sold to the public or traded on an exchange. For example, a federal government regulatory agency called the Commodity Futures Trading Commission (CFTC) approves every kind of futures contract before it sees daylight. Moreover, for a security to trade on an exchange, the issuing company must satisfy some listing requirements in terms of the company’s assets, annual earnings, shareholder interests, and audit requirements, among other factors. The NYSE has the steepest requirements among the American exchanges, and the US listing requirements are generally stricter than those in Asia or Europe. Exchanges tend to have strict rules and regulations, codes of conduct for members, and self-governance procedures. By lowering the likelihood of market manipulation and fraud, exchanges make traders comfortable, which increases trade volume (business). No exchange likes to have a government regulator at close heels, and a good self-governance program keeps them away. Moreover, self-regulation can stave 2

Most OTC transactions involve a dealer who stands ready to buy and sell securities from an inventory that she maintains. The next section discusses dealers in greater detail.

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off legal actions by angry customers who may otherwise feel cheated. Self-regulation is costly, but like a vaccination, its benefits outweigh the costs. Most exchanges also have a system of mediating disputes (arbitration) that handles problems early in the process and lowers the chances of lawsuits. Although small offenses may avoid detection, the big ones tend to get punished. Regulators and exchanges regularly report names of guilty individuals and their punishments. The press disseminates this information to the general public. OTC contracts require no regulatory approval. Organized OTC markets, being an electronically connected network of spatially separated traders, have fewer restrictions than an exchange. They offer investors greater investment choices but little transaction safety. For many OTC markets, when the going gets tough, the tough may skip town—there is risk of a counterparty failing to honor his side of the contract. In OTC markets, as in life, you need to remember the adage “know your customer.”

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3.3

Brokers, Dealers, and Traders in Securities Markets

Want to buy a house or get insurance protection? Call a broker. Need a new car or planning to purchase a fancy electronic gadget? See a dealer. Because people trade at different times, and they desire to see a collection of goods to make their pick, dealers are useful. Dealers intermediate across time by maintaining an inventory of goods. Buyers can buy from and sellers can sell to a dealer. The dealer sells a commodity at the offer or the ask price and buys the commodity at the bid price. Ask and bid prices were defined in Chapter 1. (Write a [for ask] above and b [for bid] below; Ask is the Buying price for the Customer [abc].) For example, a used car dealer sets two prices: the retail (or the ask) price at which he sells cars and a wholesale (or the bid) price at which he buys cars. The dealer earns a living from the difference between these two prices, which is known as the bid/ask (bid/offer) spread. In finance, spread has four different uses: (1) the gap between bid and ask prices of a stock or other security, (2) the simultaneous purchase and sale of separate futures or options contracts for the same commodity for delivery in different months, (3) the difference between the price at which an underwriter buys an issue from a firm and the price at which the underwriter sells it to the public, and (4) the price an issuer pays above a benchmark fixed-income yield to borrow money. These spread definitions (except for the third definition, which is more relevant in a corporate finance course) will be used throughout the book. Paucity of information concerning trading opportunities is an impediment to transactions. Brokers are intermediaries who help to overcome this hurdle and facilitate transactions. They match buyers and sellers and earn commissions for this service. Brokers have no price risk because they carry no inventory and do not trade on their own accounts. Dealers face price risks because they hold inventories. In both exchanges and OTC markets, brokers and dealers play significant roles. They earn a living by maintaining smoothly functioning, orderly markets. Many exchanges designate specialized dealers as market makers. They make markets

BROKERS, DEALERS, AND TRADERS IN SECURITIES MARKETS

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by posting ask and bid prices and stand ready to trade at those prices throughout the trading day—and enjoy enhanced trading privileges for their services. In OTC markets, some dealers take up market-making functions. Brokerage and dealership are risky business—charging too much will drive away customers, whereas charging too little will wipe the business out. Risks originate from many sources: ■

Set security/margin deposit levels. Higher margins make brokers safer, but they are costlier to customers.

■

Inventory to maintain. A larger inventory provides more to sell, but it’s costlier to maintain and makes the dealer more vulnerable to price declines.

■

Bid and ask prices to post. A narrower spread means more transactions but fewer profits per trade, while a widening of the spread has the opposite effect.

■

The amount of securities to bid or offer at those prices. A higher amount at any price exposes the dealer to a greater risk of sharp price movements.

Brokers and the dealers understand these trade-offs and finely balance risks and expected returns to survive in ruthlessly competitive markets. The setting of bid and ask prices is the most important decision that a dealer faces. There are many ways of characterizing traders in security markets. One common practice is to distinguish them as individual investors or institutional traders. For our purpose, it is more useful to categorize them in terms of their trading strategies either as arbitrageurs, hedgers, or speculators. Arbitrageurs seek price discrepancies among securities and attempt to extract riskless arbitrage profits. Hedgers try to reduce risk by trading securities and are often cited as the chief reason for the existence of derivatives markets. Speculators take calculated risks in their pursuit of profits; they may be classified as scalpers, day traders, or position traders on the basis of how long they hold their trades: 1. Scalpers trade many times a day with the hope of picking up small profits from each transaction. They are physically present on an exchange’s trading floor and pay very low trading costs. Scalpers’ holding periods are often measured in minutes and seconds. Market makers are scalpers. Scalpers make markets more liquid by standing ready to buy and sell at posted prices. Liquidity is a desirable feature of securities markets. It’s the ease with which an asset can be converted to cash, and vice versa. 2. Day traders try to profit from price movements within the day. They open their trading positions in the morning and close them out before going home at night. 3. Position traders (also called trend followers) maintain speculative trading positions for longer periods of time. Position traders often try to identify and capture the abnormal price differences between two assets (a spread), with the hope of reaping a profit when the spread is restored to its historical values. These distinctions aren’t ironclad, and traders sometimes flit from one role to another.

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3.4

Automation of Trading

Toward the turn of the century, rapid advancements in IT gave us revolutionary products and services, for example, cell phones, the Internet, and personal computers. Exchanges, brokers, and dealers exploiting these technologies have lowered trading costs and in many cases eliminated human labor from the trading process. IT has also revolutionized trading: innovations like electronic exchanges, online brokerage accounts, and network linkages among exchanges are now commonplace.

3.5

The Three-Step Process of Transacting Exchange- Traded Securities

Security trading on an organized exchange generally involves three steps: execution, clearing, and settlement. OTC transactions follow a similar approach, except that there is no clearing done by a third party (Figure 3.1 shows these three operations for a typical stock trade):

FIGURE 3.1: Execution, Clearing, and Settlement in the Stock Market A) Execution

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Buyer

Seller

Broker

Agree on trade

Broker

(Alternately, a dealer may replace a trader and her broker) B) Clearing (after market closing but before market opening on next trading day) Brokerage firm representatives meet with clearinghouse officials to “clear” (recognize and record) a trade. C) Settlement (one or more business days after execution) Buyer pays seller and gets ownership of the securities.

BUYING AND SELLING STOCKS

■

Execution. A trade is executed when the buyer (or her representative broker) and the seller (or his rep) “meet” on an exchange, physical or electronic; agree on price and quantity; and commit to trade.

■

Clearing. Before the market reopens, an executed trade must clear through the exchange’s clearinghouse. Some brokers and dealers become clearing members, who, besides clearing their own trades, clear trades for their clients and for nonclearing brokers and dealers. The clearinghouse clears a trade by matching the buy and sell orders, recognizing and recording the trade. The clearinghouse in a derivatives exchange performs the additional role of guaranteeing contract performance by becoming a seller to each buyer and a buyer to every seller.

■

Settlement. Finally, a trade ends with a cash settlement when the buyer pays for and gets the securities from the seller. In the old days, this involved the exchange of an ownership certificate for cash or check. Nowadays, it is usually done through transfer of electronic funds for ownership rights between brokerage accounts.

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IT advances have drastically reduced the time lag between these three steps. Trades can be executed in the blink of an eye and even settled on a real-time basis. While the NYSE and NASDAQ remain dominant venues for stock trading, there are more execution choices than ever before. Your broker has a “duty to seek the best execution that is reasonably available for its customers’ orders” (see www.sec.gov/investor/pubs/tradexec.htm). For example, a broker may send your order: ■

to trade an exchange-listed stock. to a national exchange (like the NYSE), to a smaller regional exchange, or to a firm called a third market maker, which stands ready to trade the stock at publicly announced prices

■

to trade an OTC stock to a market maker in the OTC markets

■

to an electronic communications network (ECN), which is an electronic system that automatically matches buy and sell orders at specified prices

■

to another division of his firm to internalize the order by filling it from the firm’s own inventory

You can also instruct the broker to route your order to a specific exchange, a particular dealer, or a predetermined ECN.

3.6

Buying and Selling Stocks

Stocks have been trading in Europe for centuries. The oldest stock exchange opened in Amsterdam in 1602 with printed shares of the United East India Company of the Netherlands. The London Stock Exchange traces its origins back to 1698, when a list of stock and commodity prices called “The Course of the Exchange and other things” was issued, and stock dealers, expelled from the Royal Exchange for rowdiness, started to operate in the streets and coffeehouses nearby (see the London Stock Exchange’s website at www. londonstockexchange. com). The nineteenth century belonged to

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England—Britannia ruled the waves, the sun never set on the British Empire, and London was the commercial capital of the world. After World War I, the hub of commercial and trading activity shifted from London to New York and to the NYSE.

Trading at the New York Stock Exchange The NYSE began in 1792, when 24 prominent stockbrokers and merchants assembled under a buttonwood tree on Wall Street in Lower Manhattan and signed an agreement to trade charging a uniform commission rate. Eventually, the Big Board (a popular name for the NYSE) became the world’s best known and one of the largest stock markets. The exchange heavily invests in technology and has pioneered many innovations such as the stock ticker, an automated quotation service, electronic ticker display boards, and electronic order display books. Example 3.1 discusses a hypothetical trade on a typical exchange.

EXAMPLE 3.1: Exchange Trading of Stocks

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Trade Orders ■

Suppose Ms. Longina Long wants to buy a round lot (one hundred shares) of YBM, which is the abbreviation for Your Beloved Machines Inc., a fictitious company that we use throughout this book. Assume that like many major US companies, YBM trades on a stock exchange. Ms. Long has opened a brokerage account and placed enough funds into it for trading. She places an order by calling up her broker or submitting the order through an Internet account.

■

The broker’s representative records and time stamps her order; web orders automatically record this information. Smaller orders are usually executed quickly. Block trades that involve ten thousand shares or more are often negotiated away from the trading floor, with or without the help of brokers.

■

Various types of orders can be submitted. A limit order must be filled at the stated or a better price or not traded at all. By contrast, a market order must be immediately transacted at the best available price. A liquid stock is actively traded and has many limit buy and sell orders around the market price. As such, reasonably sized stock positions can easily be converted into cash with a minimum loss of value.

Trade Execution ■

Suppose Long submits a market order and a counterparty is sought to take the other side of the trade. This could be another investor or a professional. Long sells 100 YBM stocks at the ask price of $100.10 per share, and the trade gets executed. Ms. Long pays $10,010 for shares plus brokerage commissions. A discount broker may charge her as little as $5, while a full-service broker will cost more.

Clearing and Settlement ■

After execution, most exchanges transmit trade records via computer to a little known but powerful organization called the National Securities Clearing Corporation (NSCC) for clearance and settlement. They are sent as locked-in transactions, which already have trade details from buyers and sellers prematched by a computer. The NSCC also “nets trades and payments among its participants,

BUYING AND SELLING STOCKS

reducing the value of securities and payments that need to be exchanged by an average of 98% each day.”3 It generally clears and settles trades on a two-business-day cycle (T + 2 basis). 3

www.dtcc.com/about/subs/nscc.php.

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Over-the-Counter Trading Over-the-counter markets, along with regional exchanges, have always provided the major venue for trading small stocks and other securities that could not be listed on the Big Board. Since the 1970s, NASDAQ has been considered the major OTC market in the US. Following the Penny Stock Reform Act of 1990, the OTC Bulletin Board (OTCBB) began operations as “a regulated quotation service that displays real-time quotes, last-sale prices, and volume information in over-the-counter (OTC) equity securities.”4 Subsequently, the OTC market in the US has shifted to the OTCBB and the Pink Quote OTCBB, which has expanded over the years, bringing greater transparency to the OTC equities market. Only market makers can quote securities in the OTCBB, and FINRA rules prevent them from collecting any fees for this service. OTCBB displays market data through vendor terminals and websites. The Pink Quote (formerly known as Pink Sheets owing to the color of the paper on which the quotes used to be printed) is an electronic system that displays quotes from broker-dealers for some of the riskiest stocks trading in the OTC markets. The SEC website cautions potential traders, “with the exception of a few foreign issuers, the companies quoted in Pink Quote tend to be closely held, extremely small and/or thinly traded. Most do not meet the minimum listing requirements for trading on a national securities exchange. . . . Many of these companies do not file periodic reports or audited financial statements with the SEC, making it very difficult for investors to find reliable, unbiased information about those companies.”5 Caveat emptor, or “buyer beware,” is the guiding principle in this market.

Alternative Trading Systems: Dark Pools and Electronic Communications Networks The technology revolution has introduced securities trading in venues that are opaque. These new trading venues are called alternative trading systems (ATS). ATS such as dark pools and electronic communications networks (ECNs) are SECapproved non-exchange-based markets for trading securities. Such nonexchange trading venues are a result of the SEC’s 1998 Regulation ATS (Regulation of Exchanges and Alternative Trading Systems). Dark pools (or dark pools of liquidity) have several features. First, they are secretive trading networks that do not send an order directly to an exchange or display 4

www.otcbb.com/aboutOTCBB/overview.stm.

5

www.sec.gov/answers/pink.htm.

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it in a limit order book. Second, an interested trader either negotiates with a potential counterparty or gets matched with one by the dark pool. Third, the market is a nearexclusive domain for institutional players with large orders. National exchanges like NYSE and the NASDAQ Stock Market also route some of their orders to dark pools. ECNs are alternate trading systems that are registered with the SEC as brokerdealers. An ECN’s participants are subscribers, which include institutional investors, broker-dealers, and market makers. Individual traders can indirectly participate by opening an account and submitting trades through a broker-dealer subscriber. ECNs primarily trade stocks and currencies. Trades are usually submitted as limit orders, which get executed when the ECN matches buy and sell orders according to some protocol. If no matching order is found, an ECN may send it to another market center for execution. Unlike a dark pool, the orders are publicly displayed to all subscribers. Founded in 1969 to provide electronic trading for institutions, Institutional Networks (Instinet) has been a pioneer in electronic trading and it became the first ECN in 1997. The NASDAQ (founded in 1971) may be considered an early ECN. ECNs were formally recognized and approved in 1998 when the SEC introduced Regulation ATS. By slashing trading fees and providing better execution, ECNs have enjoyed mind-boggling growth.

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3.7

Dollar Dividends and Dividend Yields

“All men are ready to invest their money, but most expect dividends,” observed T. S. Eliot in Choruses from the Rock in 1934. As Eliot observed, dividend payments in the form of cash (cash dividend) or additional shares (stock dividend) are a timehonored way for companies to reward their shareholders. For dividend payments, the ex-dividend date is an important day. If you buy the stock before the exdividend date, you get the share and the dividend (buying stock cum-dividend); however, if you buy it on or after this date, then you get the stock without the dividend (buying stock ex-dividend). Let us analyze the behavior of the stock price on the ex-dividend date. Suppose the cum-dividend stock price is $100 and the company pays a $2 dividend. Then, we claim that the ex-dividend stock price should be $98. Why? If it were not, profithungry traders would take advantage of any deviation, and as profits and losses occur, this trading activity eventually eliminates the discrepancy from the market price. To see how this works, suppose the ex-dividend stock price is $99 instead. Then, clever traders will buy this stock just before it goes ex-dividend. Their $100 investment will immediately become $101 (ex-dividend stock price $99 plus dividend worth $2), which they can sell and make $1 in instant profit. Conversely, if the stock price falls by more than $2, say, it falls to $97 after the stock goes ex-dividend, then the traders can “go short” (borrow and sell the stock) and buy back the stock after it goes ex-dividend, locking in $1 as instant profits. In this way, $98 is the only exdividend stock price consistent with no riskless profit opportunities. We formalize this observation as a result. (see Result 3.1).

DOLLAR DIVIDENDS AND DIVIDEND YIELDS

RESULT 3.1 The Relation between a Stock’s Price Just Before and After It Goes Ex-Dividend In a market free of riskless profit opportunities,a SCDIV = SXDIV + div

(3.1)

where SCDIV is stock price cum-dividend, SXDIV is stock price ex-dividend, and div is the amount of the dividend. a

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Later, we will call this an arbitrage opportunity. Formally, this result also depends on the stock price process having a continuous sample path (see Heath and Jarrow 1988). Of course, this result also depends on the market satisfying the standard frictionless and competitive market assumptions that are introduced in later chapters.

Dividends are paid in dollars. However, to compare dollar dividends across stocks with different valuations, one computes dividend yields. A dividend yield is the dividend expressed as a fraction of stock price. In this example, the dividend yield is (2/98) = 0.0204 or 2.04 percent. Dividing by the stock price normalizes the dollar dividend and makes a comparison possible across stocks on an apples-to-apples basis. The dividend yield is also useful when dealing with stock indexes like the Standard and Poor’s 500 Index (S&P 500), which is a weighted-price average of five hundred major US stocks (indexes are discussed in more detail in Chapter 6). Accounting for hundreds of dividends (near two thousand dividends in the case of the S&P 500) and adjusting the index’s value accordingly are daunting tasks. A simple technique for making this adjustment using the dividend yield involves the following steps: (1) add up all the dividends paid over the year and express this sum as a percentage of the index value generating a dividend yield, and call it 𝛿; (2) assume that 𝛿 is paid out over the year on a continuous basis, proportional to the level of the index; and (3) assume that the dividend gets continuously reinvested in the index so that the index grows in terms of the number of units. An initial investment of one unit in this index then becomes eᄕ units after one year and eᄕT units after T years. We illustrate this computation with Example 3.2.

EXAMPLE 3.2: Dividend Yields ■

Suppose that a fictitious stock index named “INDY Index” stands at 1,000 on January 1 and has a dividend yield of 𝛿 = 0.06 or 6 percent per year. Assume that dividends are paid m = 3 times a year, where each payment is 𝛿/m = 0.02 times INDY’s price at the dividend payment date. Let us buy one unit of the index “INDY Spot.”

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■

Four months later, INDY stands at 1,100. The dividend payment is 0.02 times this amount, which equals 22. As we reinvest this dividend, the amount of the dividend is irrelevant—all we need to know is that we have 1 × 1.02 = 1.02 units of the index from now onward.

■

Four more months later, we will get another 2.00 percent of the index level as a dividend. Because we were previously holding 1.02 units of INDY Spot, we now have 1.022 = 1.0404 units of INDY Spot. Similarly, at the end of the year, we have 1.023 = 1.061208 units of the index.

■

Next, we modify the payout frequency and assume that the dividend is paid continuously and plowed back into the portfolio after each payment. Here an investment of one unit of INDY Spot will be worth eᄕT units after T years (see Appendix A). Eight months later, we have eᄕT = e[0.06×(2/3)] = 1.040811 units of the index. This property is restated as Result 3.2.

RESULT 3.2

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Valuing an Asset Paying a Continuously Compounded Dividend Yield Consider an asset that pays a continuously compounded dividend yield of 𝛿 per year, which is reinvested back in the asset. Then, a unit investment in the asset today grows to eᄕT units after T years.

Note that Result 3.2 gives a quantity adjustment to the stock price to incorporate a dividend payment, while Result 3.1 is a price adjustment. These two modifications will be used later in the book when considering stocks paying dividends.

3.8

Short Selling Stocks

There are two ways of selling shares: selling a stock from one’s existing portfolio and selling short a share that one does not own. The first transaction is as straightforward as stock buying, whereas the second involves a more complex procedure. A short seller borrows shares from an existing stockholder and then sells them. He has the obligation to buy back the shares in the future and return them to the lender. Moreover, he owes the lender any dividends that the company pays. Interestingly, the act of short selling artificially creates more long shares than were originally issued in the primary market. Later in the book, you will see that this can become a critical factor in market-manipulation trading schemes.

SHORT SELLING STOCKS

A stock buyer is bullish because she expects that the stock will go up in value. A stock seller is bearish because he expects the stock to fall in value. A short seller is much more aggressive in his bearish stance, unlike a seller, who simply gets rid of a stock. A short seller bets on his negative view by undertaking a fairly risky trade, which could be quite dangerous if the bet goes wrong. Chapter 10 discusses an example in which a manipulator corners the market and squeezes the shorts. To see how short selling works, let’s walk through a short sell example (see Example 3.3).

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EXAMPLE 3.3: Short Selling ■

“Typically when you sell short, your brokerage firm loans you the stock. The stock you borrow comes from either the firm’s own inventory, the margin account of other brokerage firm clients, or another lender,” informs an article on the SEC’s website.7

■

Suppose Mr. Oldham owns one hundred shares of BUG (Boring Unreliable Gadgets, a fictitious company name), which are held in his account at Mr. Brokerman’s firm. To make them available for short selling, the shares are held in “street name.” Oldham doesn’t gain anything from this—he is just helping Mr. Brokerman earn some commissions. For simplicity, let’s assume that Mr. Brokerman is the only broker in town.

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Brokerman borrows one hundred shares from Oldham, lends them to Shorty Stock, and helps Shorty sell them short to Mr. Newman (see Figure 3.2). In the future, Shorty has to buy one hundred BUG shares, no matter what the price is, and give them back to Oldham. He owes any dividends paid over this period to Brokerman, who credits them to Oldham’s account.

FIGURE 3.2: Short Selling Example

Brokerman Borrows

Oldham Owns 100 BUG shares in street name; loses voting rights

Pays dividends

Newman Owns 100 BUG shares Dividends from BUG Short-sells

Shorty Stock Shorty sold 100 BUG shares short

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The market is now long two hundred and short one hundred BUG stocks. Still, from BUG’s perspective, there are only one hundred outstanding shares owned by Newman. The other one hundred shares held long by Oldham and the one hundred shares short sold by Shorty Stock are artificial creations, for they cancel one another. Newman gets dividends from BUG and has the voting rights.

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Strangely, Oldham’s original shares became artificial shares during the short selling process. They are Shorty Stock’s babies, which he has to look after. Thus Shorty has to match any cash dividends, stock dividends, and so on, that Oldham would have rightfully received.

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Dividends lower a stock’s value and benefit Shorty Stock because he can now buy them more cheaply. A dollar dividend on a $50 stock will lower its value to $49, which will help Shorty earn a dollar if he now buys back the stock. Paying Oldham this dollar will keep everyone happy by making things fair.

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The only thing that Oldham really loses is voting rights because BUG will not recognize an extra owner. He doesn’t mind that, unless he is planning to thrust major changes on BUG through proxy fights.

7

“Division of Market Regulation: Key Points about Regulation SHO” describes the short selling process for retail customers (see www.sec.gov).

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3.9

Margin—Security Deposits That Facilitate Trading

The word margin has many meanings. In economics, the terms margin and marginal are often used to denote what happens as a result of a small or unit change. In accounting, the term profit margin (of a business) refers to the amount by which revenues exceed costs. In financial markets, margin means “a sum deposited by a speculator with a broker to cover the risk of loss on a transaction on account; now esp. in to buy (also trade, etc.) on margin” (according to the Oxford English Dictionary Online). Margins and collaterals are fundamental to any security market transaction involving explicit or implied borrowing. Let’s consider some examples: ■

Buying stocks on borrowed funds. You decide to leverage (usually called gearing in Europe) your portfolio to increase your potential returns. Suppose you invest $10,000 of your own funds and take a loan of $8,000 from your broker to buy securities worth $18,000. You must open a margin account (also known as a cash and margin account) with your broker. Your investment of $10,000 is the margin or collateral, which acts like a security deposit. Now, a 10 percent return on your leveraged portfolio will be $1,800 earned on your investment, an 18 percent return. But leverage cuts both ways because losses also get magnified. Your actual return is lower because the broker charges you interest on the daily balance of your loan until you pay him back.

MARGIN—SECURITY DEPOSITS THAT FACILITATE TRADING

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67

Short selling stocks. A short seller must also open a margin account. The broker requires you to keep the proceeds from the short selling plus additional funds as margin in this account. Remember that to short, you borrowed the stock first.

Before initiating a trade that requires margin, you must put up the initial margin. This could be in the form of cash or high-quality, low-risk securities. For the broker’s protection, there is a maintenance margin requirement: the minimum amount that you must maintain in the account to keep it open. If your position loses value and the margin account balance declines to this threshold level, the broker will issue a margin call that requires you to promptly provide enough cash (maintenance margin is always in the form of cash) to restore your account balance to the maintenance margin level. If you fail to do this, the broker liquidates your portfolio. Extension 3.1 shows how margin account adjustments work.

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EXTENSION 3.1: Margin and Stock Trading As a result of the great stock market crash of 1929, the US government would like to control the level of stock speculation and has entrusted the Federal Reserve Board with the job of setting minimum margin requirements for securities trading. The exchanges and brokerage firms may set them at even higher (but not lower) levels. However, the real role of margin in financial transactions is to minimize counterparty credit risk. Currently the Fed sets the initial margin requirement at 50 percent and the maintenance margin at 25 percent for stocks (and convertible bonds). This means that when initiating a margin trade, you must come up with at least 50 percent of your own funds and finance the rest with a loan from your broker. Your purchased securities will appear with a credit sign and the loan with a debit mark in your brokerage account, and the broker will charge interest on it. As it is a secured loan (a low-risk loan backed by collateral; the market value of the stock is much more than the amount borrowed), your broker will probably charge the call loan rate (or broker’s call) and add a slight premium. Adding a spread to the benchmark libor index rate usually determines the broker loan rate. It is regularly published (along with other interest rates) in the Wall Street Journal and in the business sections of many other newspapers. The following formula is used for margin computations.

Formula 1: Margin Computation for Stock Trading The margin (or security deposit) computation in connection with buying stocks on margin (by taking a loan from a broker) or short selling is given by the formula Margin = [(Market value of assets –Loan) /Market value of assets] If maintenance margin is 25 percent, then your equity must be at least 25 percent of the portfolio value. If your equity has dropped below this level, you must respond to a margin call, deposit additional funds, and bring the equity back to the 25 percent level. Your firm may set its own margin requirements (often called “house requirements”) at a higher level, say, 40 percent, for maintenance margin. Margin adjustments can get very complicated (see Sharpe, Alexander, and Bailey 1999). The following example illustrates the computations involved (see Ext. 3.1 Ex. 1).

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EXT. 3.1 EX. 1: Margin Account Adjustments for Stock Trading

Buying Stocks on Margin ■

Suppose you buy two hundred YBM stocks at $100 each. The purchase price is $20,000, and you decide to borrow 50 percent of this amount, $10,000, from your broker. Your investment is $10,000. By our formula, initial margin is given by (20, 000 − 10, 000) /20, 000 = 0.5 or 50 percent

■

Suppose YBM goes up to $110. Then - the market value of your assets is 200 × 110 = $22,000 - the loan amount is still $10,000 - the investment in your account is (Market value of assets – Loan) = 22,000 – 10,000 = $12,000 - the margin in your account = 12,000/22,000 = 0.5455

■

Suppose YBM goes down to $60. Then - the market value of the portfolio is 200 × 60 = $12,000, and your investment is 12,000 – 10,000 = $2,000 - the margin = 2,000/12,000 = 0.1667 - you will receive a margin call to come up with more cash, an extra $1,000, so that your margin meets the 25 percent requirement

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Margin Account for a Short Seller ■

Suppose you decide to sell YBM short instead and hope to use the $20,000 in proceeds to buy a new red car. However, the broker not only refuses to let you touch any of this money but also requires you to deposit an additional 50 percent, or $10,000, into your margin account. Isn’t this outrageous?

■

If you think carefully, the broker’s actions are entirely justified. Consider what would happen if he keeps just $20,000 and the stock price goes up by just 1 cent. Your account now has a negative value of – 200 × 100.01 + 20,000 = –$2. The 50 percent initial margin requirement provides the cushion he needs. Whereas a stock buyer may or may not open a margin account, a short seller is always required to do so.

■

In contrast to a regular stock purchase, a short sale involves an initial sell and a subsequent buy. Consequently, any increase in the stock price will increase the liability and eat up margin in your account.

3.10

Summary

1. This chapter has two themes: a study of securities trading in general and stock trading in particular. Institutional details matter in finance. Stock trading is the easiest type to understand, and the trading of different derivatives can be seen as extensions to this process. 2. Stocks first trade in primary markets and then in secondary markets. 3. Exchanges were characterized by central physical locations where buyers and sellers gathered to trade. Nowadays, many exchanges are completely electronic and have a central computer executing the trades.

SUMMARY

4. Any trade away from an exchange is called an over-the-counter transaction. There are organized OTC markets (interbank markets), which are diffused networks of buyers and sellers, brought together by telecommunication connections. 5. Brokers match buyers and sellers and earn commissions for this service. Dealers trade on their own account and survive by posting a bid/ask spread. 6. Traders may be classified into three categories: hedgers who use securities for risk reduction, speculators who accept risky trades with profit expectations, and arbitrageurs who try to find price discrepancies and extract riskless arbitrage profits with strategies like buying low and selling high. Speculators can be further distinguished as scalpers, day traders, and position traders on the basis of how long they hold their trades. 7. The IT revolution has given us online brokerage accounts, electronic exchanges, and network links among exchanges and has equipped individual investors with unprecedented resources. Electronic exchanges minimize human involvement in the trading process and have eliminated floor trading altogether. 8. Security trading in an organized exchange generally involves three fundamental steps: execution, clearing, and settlement. There are more execution choices today than ever before. A trade may be sent to a national exchange, to a smaller regional exchange, to a third market maker, to a market maker in the OTC market, to an alternative trading system such as an electronic communications network, or to another division of the firm.

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9. The NYSE is a prototypical securities market. It merged with an electronic exchange and demutualized and restructured itself as a for-profit company. 10. Dividend payments are a time-honored way of rewarding stock investors. When the stock goes ex-dividend, the cum-dividend stock price equals the ex-dividend stock price plus the dividend. 11. The concept of continuously compounded interest can be used to study an investment in a portfolio that mimics the cash flows from a stock index. Consider one unit investment in an asset that pays dividends according to a continuously compounded yield 𝛿 per year, which is reinvested back in the asset (assume dividend 𝛿/m is paid and compounded m times per year, where m becomes infinitely large). Then, after T years, the portfolio will contain eᄕT units of the asset. 12. A short seller borrows shares held in street name and sells them short. The original owner loses voting rights, and the short seller compensates the owner for dividends. 13. Although the concept of margin is similar for trading different securities, the transactions vary. Traders buying stocks open a margin account only if they would like to finance a part of their purchase with a loan from the broker. Margins are security deposits that can be in the form of cash or some high-quality security. Margin account holders must start with an initial margin. When the account balance falls below the maintenance margin level, the broker issues a margin call, at which time the account holder must come up with enough cash to restore

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the account balance to the maintenance margin level. Short sellers are always required to open margin accounts and must keep the entire sale proceeds plus some additional margin with the broker.

3.11

Cases

Martingale Asset Management L. p. in 2008: 130/30 Funds and a Low Volatility Strategy (Harvard Business School Case 209047-PDF-ENG). The case discusses

the mechanics and the economic implications of leverage and short selling for investment strategies and evaluates minimum volatility stock investment strategies and quantitative investing in general. Deutsche Borse: (Harvard Business School Case 204008-PDF-ENG). The case

explores the implications of Deutsche Borse’s acquisition of a stake in a company specializing in clearing, settlement, and custody of securities across borders. ICEX: Making a Market in Iceland (Harvard Business

School Case 106038PDFENG). The case examines the impact of increased performance on the international visibility and positioning of the Icelandic Stock Exchange and considers various options for stock exchange growth in the backdrop of the country’s strong economic performance during the period.

3.12

Questions and Problems

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3.1. Explain the difference between a stock and a bond. 3.2. Explain the difference between an exchange and an OTC market. 3.3. Explain the difference between a broker and a dealer. 3.4. Explain the difference between a bid and an ask price. 3.5. Explain the difference between a market order and a limit order. 3.6. A forward market is trading for future purchase of a commodity, while a spot

market is trading for immediate purchase. Is the stock market a spot or forward market? 3.7. For the stock market, can there be a difference between the total number of

shares issued by a company and the total number of shares held by investors in the market? If yes, explain why. 3.8. Explain the difference between execution and settlement. 3.9. Explain the difference between bull and bear markets. 3.10. The SEC regulates American stock markets. However, NYSE members have

committees that carry out a host of self-regulatory activities. NYSE members are profit seeking—why would they self-regulate themselves, when regulations only raise their cost of doing business? 3.11. Why does a dealer offer to trade only a fixed amount at the bid and ask prices?

QUESTIONS AND PROBLEMS

3.12. You are a dealer and post a price of $50.00 to $50.50 for a stock. The buy

orders outweigh sell orders, and your inventory is dwindling. How should you adjust the bid and the ask prices, and why? 3.13. What is the difference between an arbitrageur, a hedger, and a speculator? 3.14. What does trading on the OTC mean? 3.15. What is the difference between a stock trading ex-dividend and cum-dividend? 3.16. Suppose that a stock pay a $5 dividend at time t. The dividend is announced

at time (t – 1), when the stock trades cum-dividend at a price of $100. What should be the ex-dividend price at time t? Explain your answer. 3.17. Explain how to sell a stock short, assuming that you do not own the underlying

stock. Why would one short sell a stock? 3.18. Consider the following data: YBM’s stock price is $100. The initial margin is

50 percent, and the maintenance margin is 25 percent. If you buy two hundred shares borrowing 50 percent ($10,000) from the broker, at what stock price will you receive a margin call? 3.19. If the stock price is $105 and the company had paid in the previous year two

quarterly dividends of $0.50 each and two more of $0.55 each, then what is the dividend yield? 3.20. Consider an asset that pays a continuously compounded dividend yield of

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𝛿 = 0.05 per year, which is reinvested back in the asset. If you invest one unit in the asset, how many units would you have after 1.5 years?

71

4 Forwards and Futures 4.1 Introduction 4.2 Forward Contracts 4.3 The Over-the-Counter Market for Trading Forwards Copyright © 2018. World Scientific Publishing Company Pte. Limited. All rights reserved.

4.4 Futures Contracts 4.5 Exchange Trading of a Futures Contract

EXTENSION 4.1 Futures Exchanges in China and India

4.6 Hedging with Forwards and Futures 4.7 Summary 4.8 Cases 4.9 Questions and Problems

FORWARD CONTRACTS

4.1

Introduction

Surprisingly although many think of derivatives as new and sophisticated securities, this is hardly the case. Forward contracts are the oldest known derivatives, tracing their origins back to India (2000 BC), to ancient Babylonia (1894–1595 BC), and even to Roman merchants trading grains with Egypt.1 Futures contracts are “the new kids on the block.” The oldest futures contracts traded in Amsterdam, Netherlands, in the middle of the sixteenth century and on the first futures exchange, the Dojima Rice Exchange in Osaka, Japan, in 1688. As such, there must be good reason why these contracts have existed for millennia to facilitate trade across time. With respect to those who believe that derivatives are new innovations, we see that their timing is only off by three or four thousand years—but what are a few thousand years among friends? We start our study with the simplest and the most basic derivatives, forwards and futures. First we introduce forward contracts that trade in the over-the-counter market. Next we discuss futures contracts. Forward and futures contracts are fraternal twins, for the two derivatives are very similar but not identical. We explore their similarities and differences, then we take a bird’s-eye view of how a company uses forward (and futures) contracts for hedging input and output price risk, a topic discussed in greater detail in Chapter 13.

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4.2

Forward Contracts

Intuitively, a forward contract locks in a price today for a future transaction—contract now, transact later is the mantra. More formally, a forward contract or a forward is a binding agreement between a buyer and a seller to trade some commodity at a fixed price at a later date. This fixed price is called the forward price (or the delivery price, usually denoted by F), and the later date is the delivery date (or the maturity date, usually referred to as time T). Forward contracts are derivatives as their values are derived from the spot price of some underlying commodity. By market convention, no money changes hands when these contracts are created. Such an exchange can only happen if both sides are happy with the terms of the contract and believe the contract is fair, that is, it has a zero value. To understand why, suppose the forward had a positive value to the buyer of the commodity. Then it must have a negative value to the seller. This means that when the seller enters the forward contract, his wealth immediately declines. He has been hoodwinked by the buyer! A rational seller would not freely enter into such a contract. A similar argument holds in reverse if the forward contract has a negative value. Consequently, if no money changes hands when the contract is created, it must have a zero value. As such, the delivery price written into the contract must be a fair price for future trading of the asset. How do we find this delivery price? It’s actually quite straight-forward. Chapters 11 and 12 will show how to find this price using some basic economic principles.

1

See Chapter 8 for an in-depth history.

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Derivatives trade on an exchange or in an over-the-counter (OTC) market. Like a magician pulling a hare out of his hat, derivatives also get created out of thin air! Indeed, they are created when one party wants to trade a particular derivative, finds another party to take the other side of the transaction, negotiates a price, and completes the deal. This observation highlights a classic characteristic of derivatives— they trade in zero net supply markets. A derivative does not exist until a trade takes place—unlike stocks, the buyer’s and seller’s positions net out, leaving the net supply of derivatives at zero. As such, trading derivatives is also a zero sum game. What one side of the contract gains, the other side loses, and vice versa. Some market jargon is useful in understanding derivative transactions. If you agree to buy at a future date, then you take a long position or you are going long. If you agree to sell at a future date, then you are taking a short position or you are going short. A forward buyer is bullish while a forward seller is bearish about the direction of the price of the underlying commodity. Consider the following example of a forward contract.

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EXAMPLE 4.1: The Life of a Forward Contract ■

Today is January 1. Ms. Longina Long longs to buy some gold at a fixed price in the middle of the year. Forwards trade in the OTC (interbank) market, where Long’s broker finds Mr. Shorty Short. Long and Short agree to trade fifty ounces of gold at a forward/delivery price of $1,000 per ounce on June 30 at a mutually convenient place (see Figure 4.1).

■

When the contract begins, its delivery price is adjusted so that the contract is executed without exchanging cash. Consequently, the delivery price is fair, and the market value of the contract is zero at the start.

■

Thus a derivative (forward contract) gets created through mutual agreement. The brokers collect commissions from the traders for their matching service.

■

If Long and Short want, they can close out their positions early through mutual agreement. Theoretically speaking, they can sell their respective sides of the trade to other investors in the secondary market. Practically speaking, it’s a hard task. Surely they can trade with others to stop price risk, but credit risk will remain. If they don’t close out their positions early, Long and Short meet on the maturity date of June 30.

■

What happens on June 30 depends on where the spot settles on that fateful day. Because they agreed on physical delivery, Short sells fifty ounces of gold to Long at $1,000 per ounce. Had they decided on cash settlement instead, the loser would have paid the price difference to the winner. If the spot price of gold is higher than the delivery price of $1,000 per ounce, then Long wins and Short loses— Long pays less for gold than it’s worth in the spot market. Conversely, if the spot price of gold is lower than $1,000 per ounce on June 30, then Long loses and Short wins—she pays more for that gold than it’s worth in the spot market. The forward contract terminates after delivery.

FORWARD CONTRACTS

FIGURE 4.1: Timeline for a Forward Contract

Now (start date) (Time 0 = January 1)

Intermediate dates

Long agrees to buy 50 Traders can close out ounces of gold on June 30, positions by making a which Short agrees to sell. reverse trade (sell if long, buy if short).

Long buys gold for $1,000 from Short. If gold price >$1,000, Long wins. If gold price F and pays F – S(T) if S(T) < F: Short’s payoff at the delivery date T = − [S (T) − F] = F − S (T)

(4.2)

With physical delivery, the short gets paid the delivery price F, a cash inflow. He surrenders to the long an asset that’s worth S(T) in the market. 2

For simplicity, we ignore market imperfections like transaction costs, taxes, and convenience yields from holding a long position in the underlying asset. We will include these market imperfections in later chapters.

FORWARD CONTRACTS

If the contract is cash settled, then the short gets F – S(T) if F > S(T); otherwise, she pays S(T) – F to the long. Notice that the long’s and short’s payoffs are exactly equal and opposite. As mentioned previously, forward contracts are zero-sum games—if you add up the two payoffs, the net result is zero.

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EXAMPLE 4.2: Payoffs to a Long and Short Forward Contract ■

We revisit Example 4.1 again, but this time using notation. Today is January 1 (date 0). The forward price on which Ms. Long and Mr. Short agree for delivery on June 30 (date T) is F(0, T) = F = $1,000 (all prices are for an ounce of gold).

■

Suppose the spot price on the delivery date T is S(T) = $1,005. Long’s forward payoff on delivery is (1,005 – 1,000) = $5 or a gain of $5. Long pays $1,000 for gold worth $1,005 in the spot market. Short’s forward payoff at time T is –(1,005 – 1,000) = − $5 or a loss of $5. The contract forces short to accept this below-market price for gold on the delivery date.

■

If, instead, S(T) is $990 on the delivery date, then Long’s payoff at delivery is (990 – 1,000) = – $10 or a loss of $10. She pays $1,000 for gold worth only $990 in the market. Short’s payoff at delivery is – (990 – 1,000) = $10 or a gain of $10. Observe that for each possible spot price S(T), the long’s and the short’s payoffs add to zero, confirming the zero-sum nature of this trade.

■

The profit and loss for long’s and short’s forward positions on the delivery date (time T) are graphed on the profit diagram in Figure 4.2. The x axis plots the spot price of gold S(T) on the delivery date, while the y axis plots the profit or loss from the forward contract.

■

The payoff to Long is a dashed straight line, making a 45 degree angle with the horizontal axis and cutting the vertical axis at a negative $1,000, which is the maximum loss. This happens when the spot (gold) is worthless, but Long still has to pay $1,000 for it. Above this, each dollar increase in the spot price cuts Long’s loss by a dollar. If gold is worth $1, then Long’s loss is 1,000 – 1 = $999. If it’s worth $800, then the loss is $200. Ms. Long and Mr. Short break even at $1,000. If the gold price ends up higher than $1,000 on the delivery date, then Long sees profits. Figure 4.2 shows that the profit potential for a long forward position is unbounded above.

■

The payoff to Short is the downward sloping dashed straight line, making a negative 45 degree angle with the horizontal axis. Considering the horizontal axis as a mirror, Short’s payoff is the mirror image of Long’s payoff. When gold is worth 0, Short makes $1,000, as worthless gold is sold for $1,000. Thus Short’s payoff line touches the vertical axis at $1,000. For each dollar increase in gold’s price, Short’s profit declines by a dollar. If gold is worth $1, then Short’s profit will be 1,000 – 1 = $999, and so on. If gold increases beyond $1,000, then Short starts losing money. Short’s maximum loss is unbounded— if gold soars, Short plunges into the depths of despair! This shows the risks of naked short selling (short selling without holding an offsetting position in the underlying asset).

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FIGURE 4.2: Profit Diagram for Forward Contract on the Delivery Date Profit Long forward

$1,000

45°

0 F = $1,000

Short forward

-$1,000

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4.3

S(T), spot price on the delivery date

The Over-the-Counter Market for Trading Forwards

The OTC derivatives market has seen tremendous growth over the last four decades. Gigantic markets exist for foreign exchange (“forex”) forwards, money market instruments, and swaps. This is a market for the “big guys”—the banks—with excellent credit ratings. As such, it is often called the interbank market. The participants trade by telephone, telex, and computer, and they tend to be located in the world’s major financial centers such as London, New York, and Tokyo. Example 4.3 shows a typical transaction in the forex (foreign exchange) forward market.

EXAMPLE 4.3: Trading Currency Forward

The Contract ■

Suppose a US company has bought a machine worth €3 million from a German manufacturer with payment due in three months. The treasurer of the US company feels that at $1.4900, the euro is attractively priced in the spot market.3 But he doesn’t know what will happen in three months time; for example, if a euro costs $1.6000, his company will pay an extra $330,000.

THE OVER-THE-COUNTER MARKET FOR TRADING FORWARDS

■

The treasurer would like to pay today, but the company is short of cash. He checks the forward market and finds that DeutscheUSA (a fictitious name), a large commercial bank, bids euros for $1.5000 and offers euros for $1.5010 in three months’ time (see Figure 4.3). He readily agrees and locks in a price of $1.5010 × 3,000,000 = $4,503,000 for the machine. This forward market trade allows the treasurer to eliminate exchange rate risk from the transaction so that the business can focus on its core competency.

The Dealer’s Hedge ■

How does DeutscheUSA protect itself? Having far-flung operations and diverse customer bases, big banks can often find another firm to take the other side of the transaction. Suppose DeutscheUSA has branches in Germany. Contacting its customer base, DeutscheUSA finds a German importer hoping to buy €3 million worth of computer parts from the United States in three months’ time. To shed foreign exchange risk, the German importer agrees to buy €3 million for $1.5000 × 3,000,000 = $4,500,000.

■

DeutscheUSA has done something cool. It has removed price risk for both the US and German importers and managed to earn a riskless spread of (1.5010 – 1.5000) × 3,000,000 = $3,000 in the process! Skillful dealers try to perfectly offset such OTC trades.

■

In reality, such perfect offsets are unlikely, and DeutscheUSA may have residual price risk after netting out all of its foreign currency transactions. Moreover, DeutscheUSA has some credit risk if any counterparty fails, and it will have to keep tab of the transactions until the deal is done. If it desires, DeutscheUSA may hedge any residual net exposure with forex futures or by entering into a forward transaction with another dealer in the forex market.

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3

The International Organization for Standardization, or the ISO, has established three-letter codes for representation of currencies and funds. ISO’s currency codes are USD for the US dollar, AUD for the Australian dollar, EUR for the euro, JPY for the Japanese yen, GBP for the UK pound sterling, etc. When no confusion arises, we will use commonly used expressions (and symbols) like dollar ($), euro (€), yen (¥), and pound sterling or pound (£).

OTC market participants may be classified as brokers, dealers, and their clients. Usual clients are banks, corporations, mutual funds, hedge funds, insurance companies, and other institutions. Many banks act as dealers and make markets in a variety of derivatives. Sometimes clients call up dealers for quotes. Typically a dealer’s sales force makes regular cold calls to potential and existing clients, trying to sell their derivatives products. Owing to competition, simple derivatives like currency forwards aren’t very lucrative. Banks’ proprietary trading desks make more profits when they can identify some special client need and create a customized product. Big players have trading or dealing rooms, where trading desks dedicated to spot, forward, and options trading are located. Forward prices are quoted in terms of a bid and ask, and because there are no specified delivery months like June and September, forward prices get quoted for delivery in one, two, three, six, or more months from the current date. Recently, owing to new financial regulations, many banks are deemphasizing proprietary trading and focusing more on their dealership businesses.

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FIGURE 4.3: Intermediary’s Role in the Forward Transaction Initially

U.S. Co. needs to buy German machines worth €3 million in three months Faces currency risk

Faces currency risk German importer needs to buy computer parts from U.S. in three months

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After DeutscheUSA Enters

U.S. Co. agrees to buy €3 millions in three months German importer needs dollars and agrees to sell €3 million in three months

$1.5010 per euro

$1.5000 per euro

Deutsche USA (dealer) Seller (Short) Buyer (Long)

Deutsche USA has perfectly hedged its book and earns $3,000 spread. It has credit risk but no price risk.

FUTURES CONTRACTS

4.4

Futures Contracts

Forward and futures contracts are fraternal twins. Both involve an exchange of cash for an asset purchased at a later date but at prices negotiated at the start of the contract. Yet their differences (explored in Table 4.2 and Figure 4.4) make it hard to label them as identical. Just as a variety of features like air bags, antilock brakes, power steering, and seat belts make a car easier and safer to drive, a number of added features make futures easier and safer to use. You may view futures as a standardized forward or a forward as a customized futures. Unlike a forward contract, a futures trades on an organized exchange, which is a regulated marketplace where buyers and sellers gather to trade a “homogeneous product.” The federal Commodity Futures Trading Commission and the industry’s National Futures Association regulate commodity exchanges in the United States. In contrast, a forward trades in the OTC market, where no regulator tells them the dos and don’ts. An exchange standardizes futures contracts for easy trading. A trader only has to tell how many contracts to buy or sell and at what price. Standardization helps create a secondary market. It reduces transaction costs and makes futures markets more liquid. A short can easily close out her position before delivery by going long the same contract with another trader, who becomes the new short, while the original long

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TABLE 4.2: Comparison of Futures and Forward Contracts Futures

Forward

a.

Largely unregulated

Regulated

b. Trades in an organized exchange

Trades in an OTC market

c.

Customized

Standardized

d. Usually liquid and has a secondary market

Illiquid and has virtually no secondary market

e.

A range of delivery dates

Usually has a single delivery date

f.

Usually closed out before maturity to avoid taking physical delivery

Usually ends in physical delivery or cash settlement

g. Guaranteed by a clearinghouse and has no credit (counterparty) risk

No such guarantee and has counterparty credit risk

h. Trading among strangers; individual’s creditworthiness is irrelevant

Forward traders know each other and usually have high credit rating or require collateral

i.

Requires margins (security deposits)

No margin requirements (but collateral may be required)

j.

Small transaction costs

Transaction costs are high

k. Settled daily

Settled at maturity

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FIGURE 4.4: A Diagram of a Futures Contract

CFTC, NFA regulate

Broker, Exchange, Clearinghouse monitor

…

…

Daily margin payments throughout contract life

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… Starting date

…

…

…

…

Intermediate dates

…

…

… Delivery/maturity period

position remains undisturbed. In fact, most futures are closed out before delivery. This happens because making physical delivery is a costly exercise, and closing out futures early avoids this expense. If, for some reason, a position remains open until maturity, then delivery must take place on one of several dates during the delivery period. The details of this delivery procedure are discussed in Chapter 9. Contrast this with a forward contract, which is privately negotiated between two traders. As such, the costs of transacting are high. Early termination requires the consent of both sides—you cannot unilaterally close out a position before delivery. Consequently, forwards are highly illiquid, have virtually no secondary market, and are usually held until maturity. Conversely, forwards are tailor-made to suit counterparties’ needs and objectives. Forwards can be designed and traded even in situations when no futures contracts are available.

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EXCHANGE TRADING OF A FUTURES CONTRACT

A futures exchange has associated with it a clearinghouse that clears executed trades and guarantees contract performance by becoming the counterparty to every trader. As such, it eliminates counterparty risk in the transaction. The clearinghouse controls its counterparty risk by requiring traders to keep a margin (security) deposit in their brokerage accounts, which are marked to market and settled at the end of each trading day. We discuss this in greater detail in Chapter 9. By contrast, a forward contract fixes a price for a future transaction that remains locked in over the contract’s life, and settlement only takes place on the delivery date. Futures are primarily designed as risk management tools. They are poor instruments for buying and selling the commodity owing to the hassle of shipping to and from delivery points in specific locations like Chicago and New York, which may ill suit most traders. The US law takes this into consideration and distinguishes contracts on how they end. If delivery is planned and regularly happens, then the contract is likely to be classified as a forward; otherwise, it is a futures. During the 1960s and the early 1970s, some brokerage firms were offering off-exchange forward contracts that were standardized like futures (see Edwards and Ma 1992). Regulators tried to bring them under their jurisdiction, but in 1974, the US Congress denied such incursions by defining forwards as deferred delivery contracts that naturally end in delivery, thereby permitting them to trade away from the exchange floor. Although institutional differences are important, forwards and futures are also different from an economic perspective because a futures contract has daily cash flows, whereas a forward only has a final cash flow. This payout feature implies that futures and forward prices are usually different, but more important, it implies that the risks from holding the contracts differ. Futures face cash flow reinvestment risk with changing interest rates, whereas forwards do not. Because interest rates are random and constantly changing, this reinvestment risk is important, and it affects both pricing and hedging considerations.

4.5

Exchange Trading of a Futures Contract

Next we sketch the trading of a futures contract, which has evolved over time to become more electronic, which efficiently serves user needs. Moreover, this will also help us to understand options exchanges, which evolved from futures exchanges.

EXAMPLE 4.4: Exchange Trading of Futures Contracts

Order Placement ■

Today is January 1. Ms. Longina Long and Mr. Shorty Short are trying to do the transaction of Example 4.1 with exchange-traded futures contracts. They both want to trade fifty ounces of gold at a fixed price on June 30.

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■

Gold futures trade on the NYMEX and the CBOT, both divisions of the CME Group, and on other exchanges. Each regular contract is for one hundred troy ounces of gold, but there are also some mini-contracts of smaller size. No gold futures contract matures in June. There are contracts for April and July, but these months may not suit Long’s buying needs or Short’s selling desire. Standardization allows quick trading, but there is a trade-off in that it reduces available choices.

■

Futures traders must first open margin accounts (security deposits in the form of cash or some acceptable securities) with a Futures Commission Merchant (FCM).4 Old-fashioned Shorty calls up his FCM and dictates a market order to sell one July gold futures at the best available price.

■

Technologically savvy Longina submits her market order through an FCM’s website. Futures traders can place their orders in many other ways besides market orders (see Example 5.8 of Chapter 5 for different order placement strategies).

Trade Execution ■

Shorty’s broker watching his computer screen sees that most trades are occurring at a little over $1,000. He senses the market and submits a limit order to sell one contract at $1,001 or higher trade. The trade is executed.

■

Long and Short pay their respective broker commissions on a round-trip basis. There is no charge when entering a position, but the full charge is due at the time the futures transaction is closed. The amount of commission varies from trader to trader. It could be as low as $10 per round trip trade for a discount broker but far more for a full-service brokerage firm.

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Clearing a Futures Trade ■

In traditional exchanges, before the market reopens, representatives of the clearing members take the trade records to the exchange’s clearinghouse. The clearinghouse checks the records, and once the records match, the clearinghouse clears the trade by recognizing and recording it.

■

The clearinghouse also guarantees contract performance by acting as a seller to every buyer and as a buyer to every seller (see Figure 4.5). The clearinghouse minimizes default risk by keeping margins for exchange members, who, in turn, keep margin for their customers.

Settling a Futures Trade ■

A stock trade is settled when a buyer pays cash and gets the stock’s ownership rights from the seller. In contrast, a futures trade has a special feature called daily settlement, by which profits and losses are paid out daily. At the end of each trading day, a futures position is marked to market by crediting or debiting the day’s gain or loss, respectively, to a trader’s margin account, an amount that equals the difference between yesterday’s and today’s futures prices. If the futures price goes up, Long wins; if it goes down, Short gains.

■

Marking-to-market resets the delivery price at maturity for outstanding futures contracts to the current futures price in freshly minted futures. This process of marking-to-market also resets the value of a futures contract to zero at the end of each trading day.

■

To understand marking-to-market, suppose Long buys one hundred ounces of gold from Short in July for $1,001 per ounce. Suppose the next day’s settlement price is $1,004. Long would screech if you tell

EXCHANGE TRADING OF A FUTURES CONTRACT

her that she now has to buy gold in July at $1,004 because she thought she locked a price of $1,001! However, she would agree to buy at $1,004 if you were to pay her the price difference between her earlier price and the new one: ($1,004 − $1,001) = $3 per ounce. To handle this payment, $3 × 100 = $300 is taken from Short’s margin account and deposited into Long’s margin account. Wouldn’t $300 earn daily interest in a bank account? And wouldn’t such daily receipts or payments happen in an unpredictable fashion? The answer to both questions is yes. As noted earlier, these observations are what differentiate futures from forward contracts. Chapter 9 explains how marking-to-market works in greater detail.

FIGURE 4.6: Role of a Clearinghouse in a Derivative Transaction 301.56........301. .......103.71..... 99.02.......95.01.

Clearinghouse

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Buyer (Long)

Seller (Short)

Buyer (Long)

Seller (Short)

Derivative trades are executed in an exchange where the buyer and seller meet through brokers. After a trade is cleared, the clearinghouse becomes a seller to the buyer and a buyer to the seller. It has counterparty credit risk but no price risk.

Closing the Contract ■

A month later, Longina sells one July gold futures to Miss Tallmadge, closes out her position, and clears her slate. Her effective selling price after all the marking-to-markets is close to $1,001. Taxes, interest charges on margin balances, and other market imperfections will make the realized price a bit different from $1,001. Tallmadge is the new long vis-à-vis Shorty’s short.

■

Shorty decides to physically deliver the gold during the contract’s delivery period in July. He delivers one hundred ounces of gold as per exchange requirements and collects an amount close to $100,100 from Tallmadge.

4

A FCM provides a one-stop service for all aspects of futures trading: it solicits trades, takes futures orders, accepts payments from customers, extends credit to clients, holds margin deposits, documents trades, and keeps track of accounts and trading records.

Futures trading is becoming increasingly popular in different parts of the world, including in countries that were previously hostile to it. Extension 4.1 discusses futures trading in some developing countries, with particular focus on China and India.

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EXTENSION 4.1: Futures Exchanges in China and India

Moving Commodities from Farms to Consumers When you read Grimm’s fairy tales about the old days, you come across farmers who keep a portion of the crop for family consumption and sell the rest in the market. Even today, this is true in many parts of the world, where farmers’ markets are held on a regular basis. But when food moves to towns and cities, you need a supply chain—a network of storage facilities, transporters, distributors, and retailers that take food from the farmer and deliver it to the consumer. Advanced economies use sophisticated technology at each stage of the supply chain to minimize costs and maintain product quality. For example, fruits and vegetables are flash frozen, sent in refrigerated trucks, and kept in temperature-controlled warehouses to extend their shelf life. Owing to this process, restaurants can offer an incredible variety of “fish, flesh, and fowl” such as Maine lobster, Alaskan king crab, Hawaiian mahimahi, and Russian caviar. And as Walmart and many other companies have demonstrated, sophisticated supply chains can significantly lower costs that in competitive markets get passed on to consumers. However, the story is different when commodities move from rural to urban areas in some of the poorest countries. Farmers sell their excess crops to a middleman, who acts as a broker or dealer. The produce goes through several middlemen before it reaches retail outlets. Significant waste can occur during repackaging and reinspection by the middlemen, and poor handling and lack of refrigeration can further diminish the quantity. The farmers get only a small fraction of the final price. True, these intermediaries serve a useful economic function, but they are relevant only because the infrastructure is primitive and sophisticated risk management tools are absent.

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Futures Trading in China Consider China, a country home to over 1.3 billion people. During the 1980s and 1990s, China became interested in developing sophisticated financial markets and institutions. Futures trading began in China in 1993. However, during the first few years, “new exchanges opened with wild abandon, and speculative volume ballooned” (Qin and Ronalds, 2005). Eventually, the Chinese authorities closed more than forty of these exchanges. The Dalian Commodity Exchange, the Shanghai Futures Exchange, the Zhengzhou Commodity Exchange, and the China Financial Futures Exchange (founded in 2006) are the only operating futures exchanges remaining in China today. These exchanges are either fully electronic or use a combination of open-outcry and electronic trading. Interestingly, an October 12, 2009, article in the Wall Street Journal titled “China Targets Commodity Prices by Stepping Into Futures Markets” reported that the leaders of Communist China are planning to use futures exchanges “to fight back” foreign suppliers who “inflate [China’s] commodity prices.” By developing the three commodities exchanges as “major players in setting world prices for metal, energy and farm commodities,” China expects to be less susceptible to exchange prices elsewhere. Futures traders joke that “China is second only to the weather in driving some commodity prices—but less predictable.” Chinese futures prices have begun affecting global prices for many key commodities. For example, Chinese demand was a major contributor to huge swings in crude oil prices during 2008–9.

Futures Trading in India At the start of the new millennium, India removed a four-decade-long ban on commodity futures trading, allowed resuscitation of moribund exchanges, and approved the opening of new ones. Soon afterward, during 2002–3, three major electronic multicommodity exchanges, the National Multi Commodity Exchange of

EXCHANGE TRADING OF A FUTURES CONTRACT

87

India Ltd., the Multi Commodity Exchange (MCX), and the National Commodity and Derivatives Exchange Limited were formed. In yet another development, September 2010 saw the opening of the United Stock Exchange (USE) of India. Backed by government and private banks as well as corporate houses, the USE currently offers trading in currency futures (on-the-spot exchange rate of Indian rupees against the dollar, euro, pound sterling, and yen) and options (on the US dollar Indian rupee spot rate) but plans to expand its offerings to include interest rate derivatives.5 Commodity futures exchanges have also been founded in many other countries, including those in Latin America, Eastern Europe, and Asia. However, Africa has been slow in adopting futures trading. 5

See www.useindia.com/genindex.php#.

In futures trading, the terms open interest and trading volume need careful distinction: trading volume is the total number of contracts traded, whereas open interest is the total number of all outstanding contracts, which may also be counted as the total number of long (or short) positions. Trading volume is a measure of a market’s liquidity: the more the volume is traded, the more active the market is. In contrast, the open interest is a measure of the market’s outstanding demand for or exposure to a particular commodity at the delivery date. Example 4.5 further explores the distinction between these two concepts.

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EXAMPLE 4.5: Volume versus Open Interest ■

Suppose that a July gold futures just becomes eligible for trading. Heather buys twenty of those contracts from Kyle. Trade records will show the following (for convenience, they are also shown in Table 4.3): - Heather is long twenty contracts. - Kyle is short twenty contracts. - Trading volume is twenty contracts. - Open interest is twenty contracts.

■

Heather decides to reduce her exposure. She sells ten contracts to Tate. As a result, - Heather is now long ten contracts. - Tate is long ten contracts. - Kyle is short twenty contracts, as before. - Trading volume rises to thirty contracts. - Open interest is twenty contracts, as before.

■

Kyle reduces his short position. He buys five contracts from Heather. Consequently, - After selling five contracts, Heather is long five contracts. - Tate is long ten contracts, as before. - Kyle is short fifteen contracts after this trade. - Trading volume, which adds up the number of trades, is thirty-five contracts. - Open interest, which is the sum total of all outstanding long positions (or short positions), is now fifteen contracts.

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TABLE 4.3: Trade Records for Gold Futures Contracts Kyle sells twenty contracts to Heather

Trader Heather

Long

Short

Trading Volume

20

Kyle

20

Open Interest 20

20

Heather sells ten contracts to Tate

Trader Heather

Long

Trading Volume

10

30

Open Interest 20

20

Kyle Tate

Short

10

Heather sells five contracts to Kyle

Trader Heather

Long 5

Kyle

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Tate

4.6

Short

Trading Volume 35

Open Interest 15

15 10

Hedging with Forwards and Futures

Forwards and futures are widely used for hedging commodity price risks at future dates. Consider a typical firm, which uses inputs to produce one or more outputs. Input and output prices are susceptible to price fluctuations. Usually, a firm tries to shed price risk for both its inputs and outputs, provided it is not costly to do so. Then it can concentrate on production; in other words, it can focus on what it does best. Obviously, it is a judgment call to decide which risks to cure and which to endure. Suppose that to reduce input price risk, a firm sets up a long hedge (or a buying hedge) by taking a long position in a forward contract. For example, a company that mines and refines gold in some remote area and uses natural gas to power its electricity generators may find it prudent to buy a forward contract on natural gas. If natural gas prices go up in the future, then the company pays more in the spot market, but the spot market loss is largely offset by the gain on the forward contract. On the flip side, if natural gas prices go down, the company will surrender some of its spot market gains through losses on the forward position. This is always the case when one employs a hedging strategy. Hedging with forwards (and futures) cuts both ways: it

SUMMARY

reduces the risk of financial loss from adverse price movements, but also takes away potential gains from favorable price changes. Alternatively, to reduce output price risk, the company can establish a short hedge (or selling hedge) by taking a short position in a forward contract. If the company sells gold forward, then it will remove or lessen the potential for loss (as well as gains) from future spot price fluctuations. If the spot goes down, then the company loses money when it sells gold in the cash market, but it profits from the forward contract. The losses and gains are reversed when the spot price goes up. Hedging is analogous to purchasing an insurance policy on the commodity’s price. It pays off when prices move in an adverse direction. But as with all insurance policies, there is a cost—the premium. If the price does not move in an adverse way, you paid for insurance that you didn’t use. This is the cost. Risk-averse individuals will often buy insurance despite this cost, and analogously, firms will often hedge their input or output price risk.

4.7

Summary

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1. The buyer and seller in a forward contract agree to trade a commodity on some later delivery date at a fixed delivery (forward) price. Forwards are zero net supply contracts. Forward trading is a zero-sum game. On the maturity date, the buyer’s (or the long’s) payoff is the spot price minus the delivery price. The seller’s (or the short’s) payoff is equal in magnitude but opposite in sign to the long’s payoff. 2. There are organized OTC (interbank) markets for trading forwards, which are dominated by banks and other institutional traders. These markets are huge. For example, foreign currency forwards attract billions of dollars of trade every day and have outstanding obligations measured in trillions of dollars. 3. The buyer and seller of a futures contract agree to trade a commodity at a fixed delivery price on the maturity date. A futures contract is similar to a forward. The long position holder in both these contracts agrees to buy the commodity, while the short agrees to sell. The forward or futures price is set so that no cash changes hands when the contract is created. This implies that the contracts have zero initial value. 4. Exchange-traded derivatives are highly regulated contracts that trade in organized exchanges. A clearinghouse clears trades and guarantees contract performance. Traders are required to keep margins (security deposits) that are adjusted daily to minimize default risk. Exchanges also act as secondary markets where traders can take their profits or cut their losses and exit the market. 5. Many firms reduce price risk by trading futures and forward contracts. A firm can lower input price risk by setting up a long hedge buying a forward and reduce output price risk by establishing a short hedge selling a forward.

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4.8

Cases

The Dojima Rice Market and the Origins of Futures trading (Harvard Business

School Case 709044-PDF-ENG). The case blends business history with policy issues surrounding the introduction of rice futures at the Dojima Exchange, the world’s first organized (but unsanctioned) futures market. United Grain Growers Ltd. (A) (Harvard Business School Case 201015-PDFENG).

The case considers how a Canadian grain distributor can identify and manage various risks. ITC eChoupal Initiative (Harvard Business School Case 297014-PDF-ENG). The

case discusses the use of Internet technologies and derivative contracts to help poor farmers in rural India.

4.9

Questions and Problems

4.1. Define a forward contract. If a forward contract on gold is negotiated at a

forward price of $1,487 per ounce, what would be the payoff on the maturity date to the buyer if the gold price is $1,518 per ounce and to the seller if the gold price is $1,612 per ounce? 4.2. Define a futures contract. 4.3. Discuss the similarities and differences between forward and futures contracts. 4.4. What are the costs and benefits to a corn grower trading a forward contract? If

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she is expecting a harvest in three months, should she buy or sell the derivative? 4.5. Which contract has more counterparty risk, a forward contract or a futures

contract? Explain your answer. 4.6. Discuss the benefits of standardization of a futures contract. 4.7. What are the two roles that a clearinghouse plays in the case of a futures

contract? 4.8. Ang, Bong, Chong, and Dong are trading futures contracts. Carefully identify

the trading volume and open interest for each trader from the following transactions: a. Ang buys five October silver futures from Bong. b. Chong sells nine December silver futures to Bong. c. Dong buys ten October silver futures from Chong. d. Bong sells five December silver futures to Ang. e. Ang sells three October silver futures to Chong. 4.9. Suppose you are a trader specializing in futures on corn, wheat, oats, barley,

and other agricultural commodities. From the following list, which risks do you face (mark each with a yes or no): credit risk, market risk, liquidity risk, settlement risk, operational risk, legal risk.

QUESTIONS AND PROBLEMS

4.10. For forward and futures contracts, what is the difference between physical

delivery and cash settlement? 4.11. Why does a futures contract have zero value when it is first written? 4.12. What is marking-to-market for a futures? Why is this marking-to-market

important for reducing counterparty risk? 4.13. Explain why a futures contract is a zero-sum game between the long and short

positions. 4.14. Are forward contracts new to financial markets? Explain. 4.15. Can you think of a reason why forward prices and futures prices on otherwise

identical forward and futures contracts might not be equal? 4.16. What is the OTC market for trading derivatives? How do OTC markets differ

from exchanges? 4.17. When holding a futures contract long, if you do not want to take delivery of

the underlying asset, what transaction must you perform? Explain. 4.18. If you are short a futures contract, why do you not have to borrow the futures

contract from a third party to do the short sale? 4.19. Is the futures price equal to the value of the futures contract? If not, then what

is the value of a forward contract when it is written? Explain. 4.20. Is the forward price equal to the value of a forward contract? If not, then what

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is the value of a forward contract when it is written? Explain.

91

5 Options

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5.1 Introduction

Price Bounds for American Puts

5.2 Options

5.5 Exchange-Traded Options

5.3 Call Options Call Payoffs and Profit Diagrams

5.6 Longs and Shorts in Different Markets

The Call’s Intrinsic and Time Values

5.7 Order Placement Strategies

Price Bounds for American Calls

5.8 Summary

5.4 Put Options Put Payoffs and Profit Diagrams The Put’s Intrinsic and Time Values

5.9 Cases 5.10 Questions and Problems

OPTIONS

5.1

Introduction

After introducing options in an intermediate finance class, a professor asked, “Is auto insurance an option?” A serious student from the front row replied, “It is not an option because state law requires it, but you have the option to choose from different insurance companies.” Though the student was entirely correct, this was not the answer for which the professor was hoping. He took a deep breath and went on to explain that put options are very similar to insurance contracts because they restore an asset’s value after a decline and that much of option jargon comes from the insurance industry—and he resolved to frame his questions more carefully in the future. As the example illustrates, the word option has many meanings in daily life. Exchange-traded options are well-defined contracts with specific features. This chapter describes these option features and discusses how they trade. At the chapter’s end, we discuss how traders buy or sell options for various strategic objectives.

5.2

Options

Options come in two basic types: calls and puts, names you have heard before. A call option gives the buyer: - the right, but not the obligation - to buy a specified quantity of a financial or real asset from the seller - on or before a fixed future expiration date

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- by paying an exercise price agreed on today. The buyer is also called the holder or the owner. The seller is known as the grantor, the issuer, or the writer. The known future expiration date is also called the maturity date, and the fixed price is known as the exercise price or strike price. The holder exercises the call when she exercises her right and buys the underlying asset by paying the exercise price; otherwise, she lets the call expire worthless. By contrast, a put option gives the right to sell. Specifically, a put option gives the owner the right to sell the asset to the writer at the strike price until the expiration date. A buyer exercises the put when he sells the underlying asset and receives the strike price from the put writer; otherwise, he can let the put expire worthless. Options do not come free—the option buyer must pay the writer a fee (option’s price or premium) for selling those rights. The terms call and put come from what the buyer can do with these options. A call gives the option to buy, that is, to call the asset away from the writer. Conversely, a put gives the option to sell or to put the asset to the writer. In each case, the writer stands ready to take the other side of the buyer’s decision. We say that a call buyer is bullish because she expects the underlying asset to go up in value, and a put buyer is bearish because he expects the asset price to decline. Naturally, their counterparties hold opposite market views: the call writer is bearish and the put writer is bullish. We also say that the option (call or put) buyer is long the option, so the writer ends up with a short position.

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CHAPTER 5: OPTIONS

Notice that calls and puts have a lot in common with insurance policies and may be viewed as insurance contracts for hedging or risk reduction. If someone hits a car, the insurance company pays the owner enough money (subject to some deductibles) to restore the car to its original value. Likewise, a put pays the holder in case of a meltdown in the asset’s value. A call buyer benefits from an asset price rise but does not suffer from its decline. The call buyer pays for insurance to avoid losses if the stock falls. The call writer provides this insurance. For these reasons, an option price is called a premium. A premium is what you pay for the insurance. Calls and puts are examples of plain vanilla or standard derivatives. Later we will consider nonstandard or exotic derivatives, which are derivatives with more complex payoffs. Plain vanilla options are usually defined in one of two ways: American or European. A European option can only be exercised at the maturity date of the option, whereas an American option can be exercised at any time up to and including its expiration date. What we defined earlier were calls and puts of the American type. The adjectives European and American tell how the options differ in their exercise choices and have nothing to do with where they trade—in Europe, America, Africa, Asia, Australia, or even in Antarctica. European-style options trade in the US and across the Atlantic, and American-style options transact in Europe. Both American and European options have the same value on the expiration date, if the American option remains unexercised. Before expiration, however, American options are at least as valuable as their European counterparts. They duplicate what European options can do, but they also offer an early exercise feature, which is more. You don’t pay less when you get more. Chapter 6 will show how the assumption of no arbitrage makes this happen. Such simple insights are useful when trading options. The important question studied in later chapters is, how do you find an option’s premium? This has bewildered academics for a long time. A breakthrough came in 1973 with the development of the seminal Black–Scholes– Merton (BSM) option pricing model (see Chapter 19). The importance of the BSM model can hardly be overstated—it helped spawn the entire field of derivative pricing that we see today. However, option pricing still remains a Herculean task. For example, there is no generalized closed-form solution for finding an American option’s premium.1 This is important because most exchange-traded options are of the American type, and we need to price them. When closed-form solutions are not available, numerical procedures (computer programs) are used instead (see Wilmott 1998; Duffie 2001; Glasserman 2003). There remain many open questions, and option pricing remains an intellectually challenging field to which interesting contributions can still be made.

A closed-form solution is an exact mathematical formula like x = (a + b)2 . If you know a and b, you can easily solve for x. The BSM model solves an option’s price in terms of known variables and functions like the cumulative normal distribution.

1

CALL OPTIONS

5.3

Call Options

Call Payoffs and Profit Diagrams Option payoffs on the expiration date or at early exercise are easier to understand in a perfect market, where market imperfections like transaction costs, taxes, and trading restrictions are assumed away. In reality, the market contains such imperfections. The brokers who match option trades charge commissions for their services. A dealer will quote an ask price (say, $4) and a bid (say, $3.90) for the option premium. Large trades tend to have market impact, and the entire order may not be filled at the price posted by a dealer. Unfortunately, these real-world features detract from understanding the basic payoffs. Hence we study option payoffs under a perfect market structure. Once you know how these contracts work, you can adjust for these imperfections. Example 5.1 considers a call buyer’s and writer’s payoffs and profit diagrams on the expiration date.

EXAMPLE 5.1: European Call Option Payoff and Profit Diagrams

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Call Buyer’s Payoff ■

Today is January 1 (time 0). An April 100 European call option on the stock of Your Beloved Machines Inc. (YBM) is created when Ms. Longina Long pays the premium c = $4 to Mr. Shorty Short, who becomes the writer. This call has a strike price of K = $100, and it ceases trading on the third Friday of April. This is the only day this European call can be exercised (time T); see Figure 5.1 for this option’s timeline.2

■

Long’s exercise decision depends on the stock price S(T) on the fateful expiration date. If S(T) is $106, then Long exercises the call because the stock is worth more than the strike (traders say that the option is in-the-money). She pays the strike, receives the stock worth S(T), and makes [S(T) – K] = $6 in the process. The $4 premium is a sunk cost and does not affect the exercise decision.

■

Conversely, if the stock is worth less than the strike, then the option expires worthless—out-of-themoney, in traders’ parlance. If Long is keen to acquire the stock, she should directly tap the stock market. It’s imprudent to exercise out-of-the-money options. For example, if S(T) is $92, she gets 92 – 100 = –$8 by exercising the call, an outcome she should avoid. And when S(T) equals the strike price, the option is at-the-money, and the long is indifferent regarding exercise. In reality, a comparison of transaction costs for purchasing the stock is likely to influence her decision. For example, suppose Long is planning to purchase a thinly traded stock. If the stock’s spread is too high, Long may exercise an at-the-money call (or even a call that is slightly out-of-the-money) to more cheaply acquire the stock.

■

True to the proverb “a picture says a thousand words,” patterns emerge when we graph these payoffs in a payoff diagram. These diagrams plot the stock price at expiration, S(T), along the horizontal axis and the option payoff (or gross payoff ) along the vertical axis. Long call’s payoff starts at 0 value when the stock price is zero and lies flat along the axis until the stock price reaches K = $100 (see Figure 5.2). Beyond this, the payoff increases at a 45 degree angle, rising by a dollar for each dollar increase in S(T).

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Call Writer’s Payoff ■

Because option trading is a zero-sum game, Short’s call’s payoff is the mirror image across the horizontal axis of Long’s payoff. The writer loses nothing at expiration until S(T) reaches $100, beyond which his payoff declines at a negative 45 degree angle. When S(T) = $106, Short loses $6, which is Long’s gain. When S(T) = $92, Short loses nothing.

Call Buyer’s Profit ■

Payoff diagrams fail to reveal why Short, who has a zero or negative payoff on the expiration date, will get into this venture. This limitation is overcome by a profit diagram, which adds the option’s premium to the trader’s payoff. The profit diagram plots S(T) along the horizontal axis but has profit (or loss) from option trading along the vertical axis. For convenience, we ignore any interest that the option premiums may earn in the margin account. This allows us to combine the cash flows at the option’s initiation and expiration dates on the same diagram.

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FIGURE 5.1: Timeline for a Call Option

Today (start date) (Time 0)

Intermediate dates

Buyer (Long) pays the Long can exercise an call premium and gets American option. the call from the writer (Short) (received as an electronic account). Traders can close out their positions. (If none of these happens, traders meet at time T).

Expiration/maturity date (Time T)

If S(T) > K, long exercises: Long gives up the call, pays the strike price, and gets the stock from the writer. Long nets S(T) – K. If S(T) < – K, call expires worthless.

CALL OPTIONS

■

If S(T) does not exceed the strike price K = $100, Long loses c = $4, the premium she paid for the call. This is her maximum loss. When S(T) exceeds $100, Long will definitely exercise the call. A dollar increase in S(T) raises the value of her position by a dollar and cuts her losses. Consequently, the profit graph is a horizontal straight line at –$4 until it reaches the strike price, where it increases at a 45 degree angle. When S(T) = $104, Long’s $4 gain exactly offsets her initial $4 investment and gives “zero profit.” When S(T) goes beyond $104, it’s even sweeter. Technically speaking, the call holder has limitless profit potential (see Figure 5.2). Focusing on our familiar points, when S(T) = $106, Long has a net gain of $2 because she makes $6 but she paid $4 for the call. Conversely, when S(T) = $92, Long does not exercise and loses only the $4 premium.

FIGURE 5.2: Payoff and Profit Diagrams for an April 100 Call Option on the Expiration Date Payoff

Payoff

Long call 45˚

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0

100

S(T), stock price at expiration

Profit

K

0

–45˚

S(T)

–45˚

S(T)

Short call

Profit

Buy call

0 –4

45˚

100 c+K

S(T)

c 0

104 K

Sell call

Call Writer’s Profit ■

When S(T) does not exceed the strike, the call writer keeps the $4 premium collected up front. This is also the maximum profit. When S(T) increases above $100, Long’s exercise reduces this amount. Zero profits occur when S(T) = $104. Below this level, Short plunges deeper into losses (see Figure 5.2).

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In the example considered, when S(T) = $106, the writer has a loss of $2. When S(T) = $92, Long does not exercise, and Short’s profit is $4. Because option trading is a zero-sum game, the total gain of both traders is zero. In reality, brokerage commission lowers the profits accruing to each trader. ■

Notice what the diagrams reveal: call writing can generate far larger losses than call buying. The maximum loss to the buyer is the premium. In contrast, the writer can literally fall into a bottomless abyss where the maximum loss is unbounded. Recognizing this asymmetry, the options exchange requires a call writer to hold more funds in a margin account than a buyer.

2

The OCC (www.optionsclearing.com/about/publications/expiration-calendar-2010.jsp) and exchange websites label the following Saturday as the expiration date. This is a technicality. The third Friday, which we label as time (date) T, is the relevant economic date for our purpose because it is (1) the day when the option stops trading, (2) the only day a European option can be exercised, and (3) the last day an American option may be exercised. For simplicity, we will use the terms exercise date and expiration date interchangeably.

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The Call’s Intrinsic and Time Values An American call is identical to a European call, except that it can be exercised any time until expiration. Remember that if an American call is not exercised early, then it has the same value as a European call on the expiration date so that the payoff diagrams are identical. And prior to expiration, the exercise strategy for an American option is guided by two facts: (1) one should never exercise an out-of-the-money option, and (2) when exercising an in-the-money call early, one receives the stock price minus the strike price. In other words, if an American call is exercised early, then its payoff looks like that of a European call on the expiration date. We summarize these insights in Result 5.1.

RESULT 5.1 Intrinsic (or Exercise) Value of a Call Option A call holder’s payoff at expiration or if exercised early is Call’s intrinsic value ≡

S − K for S > K {0 otherwise

(5.1)

which may be written as Call’s intrinsic value = max (S − K, 0)

(5.2)

where S is the stock price on the exercise date, K is the strike price, and max means the maximum of the two arguments that follow. This is a call option’s boundary condition. The payoff is also referred to as the call’s intrinsic or exercise value.

CALL OPTIONS

What does this mean? For S < K, the term (S – K) is negative, the call is not exercised, and 0 is the value. When S ≥ K, the call’s value equals S – K. The profit is obtained by subtracting the call’s premium from the preceding payoff. This result has a simple corollary. It implies that if the stock price hits zero, then the call price is also zero. Indeed, if the stock hits zero, it has no future, and it is never going to take a positive value again. Consequently, a call, which gives the right to purchase the stock in the future, must also be zero. The intrinsic value is only part of an option’s premium. The remainder is the additional value accruing to an option because the stock price may increase further before maturity. Consequently, we define the time value of a European or American option as Time value of an option = Option price − Option’s intrinsic value

(5.3)

Time value declines as one approaches the expiration date and (by its definition) becomes zero when the option matures. Example 5.2 computes the time value of a call option.

EXAMPLE 5.2: Time Value of a Call Option

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■

On some date t before expiration, suppose that YBM’s stock price S(t) = $101 and the call price c(t) = $2.50. If the strike price K = $100, then Call option’s time value = Call price − Call’s intrinsic value = c (t) − [S (t) − K] = 2.50 − (101 − 100) = $1.50

Price Bounds for American Calls Although we are yet to price options, the boundary condition and some simple reasoning can determine what call option prices are admissible and what prices aren’t (see Example 5.3).

EXAMPLE 5.3: Price Bounds for an American Call Option ■

As in Example 5.1, consider an American call option with a strike price K = $100. Its price can never exceed the underlying stock’s price. Indeed, the call is a security that gives the holder the right to buy the underlying stock by paying the strike price. If you have to choose between the two—a gift of the stock versus getting the stock by paying some money—which one would you prefer? Getting the gift, of course, which implies that the stock is worth more than the option.

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Consequently, a call’s price can never exceed the underlying stock’s price. Figure 5.3 plots today’s stock price S(t), written as S for simplicity, along the horizontal axis and has both the stock and the call price along the vertical axis. The upward sloping 45 degree line from the origin represents the stock price. The call price can never exceed this value, and the 45 degree line forms an upper bound. A rational investor will never let a call’s price fall below the boundary condition given by Result 5.1, max(S – 100, 0)max(S – 100, 0), which forms a lower bound. This is the kinked line that starts at the origin and increases at a 45 degree angle when the stock price is $100.

FIGURE 5.3: Price Bounds for an American Call Prices Upper bound Lower bound

Call prices lie here 45˚

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0

■

K = $100

S, stock price

Let us illustrate these bounds with some numbers: - When the call is out-of-the-money, say, the stock price is $90, the time value prevents the call price from being zero. So it will lie above the horizontal axis. However, the call price can never exceed the stock price of $90, and hence it will lie below the 45 degree line emanating from the origin. - When the call is in-the-money, say, the stock price is $115, the call is worth at least S – K = 115 – 100 = $15 if exercised. So the call price can never fall below the lower bound, and of course, it can never exceed $115, which lies on the upper bound.

5.4

Put Options

Put Payoffs and Profit Diagrams We illustrate a put option’s payoff and profit diagrams with Example 5.4.

PUT OPTIONS

EXAMPLE 5.4: European Put Option Payoff and Profit Diagrams

Put Option Payoffs ■

■

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■

Today is January 1 (time 0). Ms. Long pays Mr. Short p = $6 and buys a May 50 European put option on Boring Unreliable Gadgets Corp (BUG). This put gives Long the right to sell BUG stocks on the exercise date, the third Friday of May (time T), at the strike price K = $50. A put’s timeline is similar to that for the call given in Figure 5.1. If the put is not closed out early, the traders meet on the expiration date T. If the stock price is less than the strike price (the put is inthe-money), Long exercises the put, receives K dollars, surrenders the stock worth S(T), and makes [K – S(T)] in the process. For example, if S(T) = $40, then Long sells the stock for $50 and makes $10. Conversely, if S(T) is greater than K (the put is out-of-the-money), it’s unwise to exercise. For example, if S(T) = $55, Long gets (50 – 55) = – $5 if she exercises and zero if she doesn’t. Exercise is irrelevant if BUG closes at $50 (the put is at-the-money).3 These considerations enable us to draw the payoff diagrams. For large stock prices, the put holder’s payoff has zero value, and it lies on the horizontal axis. As the stock price declines and dips below the strike, the payoff forms a kink at K and increases at a 45 degree angle, increasing by a dollar for each dollar decrease in the stock price (see Figure 5.4). The profit peaks when the stock is worthless, but Long can sell it for K dollars as per contract terms.

FIGURE 5.4: Payoff and Profit Diagrams for a May 50 Put Option on the Expiration Date Payoff

Payoff

50 Long put 0

45˚ 50

S(T), stock price at expiration

0

K –45˚

S(T)

Short put –50

Profit

Profit

44 Buy put 0 –6

45˚

6 0

K S(T)

44 –45˚ Sell put

–44

S(T)

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Because option trading is a zero-sum game, the put writer’s payoff is the mirror image across the xaxis of the buyer’s payoff. When S(T) is greater than $50, the put seller loses nothing, and the payoff coincides with the horizontal axis. When the stock goes below the strike, the payoff decreases below the horizon line at a 45 degree angle starting at K, and it reaches the nadir when the stock becomes worthless.

Put Option Profit ■

■

■

The profit diagram adjusts the payoff diagram for an option’s premium. If S(T) is greater than the strike price K = $50, Long loses the premium p = $6, which is the most she can lose. When S(T) falls below $50, Long exercises the put—each dollar decrease in S(T) raises the value of her position by a dollar and cuts her losses. Consequently, the profit graph is a horizontal line at – $6 for high values of S(T). It increases upward at a 45 degree angle when S(T) touches K = $50 on its downward journey, and it cuts the horizontal axis at the zero-profit point when S(T) = $44. It hits the vertical axis and makes an intercept of $44 when S(T) = 0 (see Figure 5.4). The profit graph for a put writer is the mirror image of the buyer’s graph across the horizontal axis. Notice that both the maximum profit and the biggest loss are bounded for a put option. Next we illustrate these profit (or loss) computations. For S(T) = $40, Long makes (50 – 40) = $10 by exercising the put. However, he paid $6 for the option, netting a profit of 10 – 6 = $4, which is also Short’s loss. For S(T) = $55, Long does not exercise the put, and her loss is the $6 premium, which is also Short’s profit.

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3

Transaction and liquidity costs may slightly modify this decision. If the long has a substantial holding in the underlying stock that is thinly traded and wants to sell, then she may exercise an at-the-money or slightly out-of-the-money put option to sell the stock at minimum transaction cost.

The Put’s Intrinsic and Time Values An American put may also be exercised before expiration. Of course, if an American put is not exercised early, then the American and European puts have identical value on the expiration date so that their payoff diagrams are identical. Before expiration, the exercise strategy for an American option is determined by two facts: (1) one should never exercise an out-of-the-money option, and (2) in case of early exercise of an in-the-money put, the value is the strike price minus the stock price. When exercised early, an American put’s payoff looks like that of a European put on the expiration date.

PUT OPTIONS

These observations are summarized as Result 5.2.

RESULT 5.2 Intrinsic (or Exercise) Value of a Put Option A put holder’s payoff at expiration or if exercised is Put’s intrinsic value ≡

K − S for S < K {0 otherwise

(5.4)

which may be written as Put’s intrinsic value = max (K − S, 0)

(5.5)

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where K is the strike price, S is the stock price on the exercise or expiration date, and max means the maximum of the two arguments that follow. This is a put option’s boundary condition. The payoff is also referred to as the put’s intrinsic (or exercise) value.

For example, consider a put with a strike price of K = $50. When S(T) = $55, K – S(T) = – $5. As the maximum of –$5 and 0 is clearly 0, the put value p(T) = 0 at expiration. When S(T) = $40, then K – S(T) = $10 is the intrinsic value. For S(T) < K, the positive term [K – S(T)] is the put’s payoff. When S(T) ≥ K, the term K – S(T) is no longer positive, and zero becomes the put’s intrinsic value. Example 5.5 illustrates the computation of a put’s time value.

EXAMPLE 5.5: Time Value of a Put Option ■

At time t, let BUG’s stock price S(t) = $48 and its put price p(t) = $3 for a strike price K = $50. Then Put option’s time value = Put premium − Put’s intrinsic value = p (t) − [K − S (t)] = 3 − (50 − 48) = $1

Price Bounds for American Puts As with calls, we can use the boundary condition to determine price bounds for put options (see Example 5.6).

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EXAMPLE 5.6: Price Bounds for an American Put Option ■

As in Example 5.4, consider an American put option with a strike price of K = $50. Its value can never exceed the strike price because the put gives the holder the right to sell the underlying stock at the strike price. Thus the maximum payoff from the put is the exercise price K = $50. This forms an upper bound for the put price and is depicted by the horizontal line at $50 in Figure 5.5.

■

The put’s price can never fall below max(0, 50 – S), the boundary condition given by (5.3), which forms a lower bound. This is the line that originates at $50 on the vertical axis, decreases at a 45 degree angle until it hits $50 on the horizontal axis, and then continues along the horizontal axis.

■

We illustrate these bounds with an example (see Figure 5.5): - When the put is in-the-money, say, the stock price is $10, the put price cannot fall below its exercise value 50 – 10 = $40. Because the put price cannot rise above its upper bound of $50, it lies between $40 and $50. - When the put is out-of-the-money, say, the stock price is $60, the put price cannot fall below the horizontal axis because no one would exercise and lose money. Of course, it cannot exceed its upper bound: the flat line at $50 above the horizontal axis.

FIGURE 5.5: PRICE BOUNDS FOR AN AMERICAN PUT

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Prices

$50

Upper bound Put prices lie here

0

Lower bound K = $50

Put option prices cannot lie in the shaded regions above the upper bound or below the lower bound.

S, stock price

EXCHANGE-TRADED OPTIONS

5.5

Exchange-Traded Options

Relative to futures contracts, exchange-traded options are newcomers. They started trading in 1973 with the opening of the Chicago Board Options Exchange (CBOE). Today options trading is dominated by big exchange groups, which offer diverse product lines (stocks, bonds, options, futures, etc.). The Options Clearing Corporation (OCC) clears the bulk of exchange-traded options in the US. As such, the OCC becomes the guarantor of contract performance to each buyer and seller. As discussed before, a clearinghouse makes derivatives safer to trade, helps develop a secondary market, allows anonymous trading, and eliminates the need to track one’s counterparty. These benefits have greatly contributed to option market growth. Example 5.7 considers a hypothetical options trade.

EXAMPLE 5.7: Exchange Trading of a Call Option

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Order Placement ■

Today is January 1 (time 0). Ms. Longina Long and Mr. Shorty Short have opened margin accounts and have been approved for trading options with their respective brokers. Unbeknownst to each another, they simultaneously decide to trade ten April 100 call options on Your Beloved Machines Inc. (YBM). These American calls have a strike price of K = $100 and an exercise date of the third Friday of April. Because each option contract is on one hundred shares, ten contracts reflect gains or losses from one thousand shares.

■

Tech-savvy Longina flips open her laptop and logs onto her brokerage account. Finding that YBM April calls have a bid price of $4.00 and an ask–offer price of $4.05, she types in a limit order that instructs her broker to buy ten contracts at $4.00 or a lower price. Old-fashioned Shorty calls up his broker and places a market order that is immediately executed. Small orders are matched via computers and are filled and confirmed within a few seconds. Most option trades are executed on the exchange floor.

■

Suppose Long’s and Short’s orders are sent via computer to their respective brokerage firms.

Trade Execution, Clearing, and Settlement ■

Shorty’s broker accepts Longina’s limit order, and ten contracts are executed at c = $4.00. Longina sees the trade confirmation on her web page, and Shorty gets a confirmatory phone call from the broker’s representative. Vital transaction data also go into the exchange’s main computer system.

■

Commission rates vary. A discount broker typically charges a fixed rate, which may be as small as $5 to $20 per trade, and a variable rate, which typically ranges from 50 cents to $2 for each option on one hundred shares. Many firms charge lower commissions for (1) internet orders; (2) active traders; (3) the simplest orders like market orders; (4) small-sized orders, because they are unlikely to be based on privileged information; and paradoxically, (5) larger orders, to give customers a break. Brokers charge two commissions: one at the time of buying (or selling) and a second commission when selling (or buying back) or exercising the option.

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Suppose Long’s discount broker charges her $10 for the trade plus a dollar per contract—$20 in all. Consequently, Long’s brokerage account is debited an amount equal to the buying price plus commission: Price × 100 × (Number of contracts) + Commission = 4 × 100 × 10 + 20 = $4, 020

■

Shorty has a full-service broker who charges $50 for the trade plus $3 per contract—$80 in all. Short’s margin account is credited an amount equal to the selling price less the commission: Price × 100 × (Number of contracts) − Commission = 4 × 100 × 10 − 80 = $3, 920

Because selling options is a risky transaction, Short’s broker requires him to keep the entire proceeds as well as some extra funds as a security deposit in the margin account.

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Closing the Contract ■

Being exchange-traded contracts, options have secondary markets. Traders retain the flexibility of taking their profits or cutting their losses and walking away. Long may sell the option to a third party and close out her position—she may win or lose in the process. Quite independently, Short can buy the option in a closing transaction, and a third trader then becomes the writer. Because this is an American call, Long can exercise at any time until the maturity date.

■

Fluctuations in option prices affect the margin account’s value. Suppose that YBM rallies tomorrow and the call price increases by $3. Then Long’s margin account gains $3,000, and Short’s margin account declines by the same amount. If his maintenance margin is breeched, Shorty has to add funds to his margin account.

■

Suppose that the option is neither exercised nor closed out early so that the counterparties meet again on the expiration date. To avoid the hassle of paying the broker a fee for exercising and yet another fee for trading YBM shares, they decide to close out their positions just before the option’s expiration and trade at a $7 premium per option. Then Long’s profits are Sale proceeds − Cost = (7 × 100 × 10 − 20) − 4, 020 = $2, 960 and Short’s profit (remember, his brokerage costs are higher) is Sale proceeds − Cost = 3, 920 − (7 × 100 × 10 + 80) = − $3, 160

ORDER PLACEMENT STRATEGIES

5.6

Longs and Shorts in Different Markets

Market practitioners often use the terms long and short to convey the notions of buying and selling. But these terms have different meanings in different markets. For derivatives, long means “buy,” and short means “sell.” Derivatives trade in zero net supply markets, where each buyer has a matching seller. Long and short in the derivatives markets are just opposite sides of the same trade. Consequently, it is a zero-sum game because one trader’s gain is the other’s loss. The short seller in the stock market does something more complex. She borrows shares, sells them, and buys back those shares at a later date to return them to the lender. For futures and forwards, both a long and a short have obligations to transact at a future date. The long agrees to buy the asset, and the short agrees to sell. Options give the buyer the right to buy (call) or sell (put) at a fixed price until a fixed date. No matter whether it’s a call or a put, the buyer who is long has an option. They decide to exercise or not. In contrast, the writer, who holds a short position, has an obligation. The short needs to fulfill the option’s terms as chosen by the long. For this situation, the buyer pays the writer a premium at the start.

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5.7

Order Placement Strategies

All traders face the same dilemma: to trade immediately or to wait to see if one can get a better price. If you want to transact immediately, submit a market order; however, if you can wait for a better price, then submit contingent orders that depend on the price, time, and other relevant factors. But there is a trade-off: market orders may get traded at a bad price, whereas price-contingent orders may never get filled. Example 5.8 considers different types of order placement strategies for trading stocks in fairy-tale (hypothetical) situations. These same order placement strategies also apply to futures and options. Let’s start our discussion with stocks for easy illustration and then show how the argument extends to derivatives like options.

EXAMPLE 5.8: Order Placement Strategies

A Market Order ■

A dealer quotes an ask–offer price of $100.10 and a bid price of $100.00 per share for YBM stock. Hoping that it will rally, you want to buy YBM in a hurry. You can place a market buy order, which gets executed immediately at the best price your broker can find—very likely at $100.10. You may be forced to pay a bit more or enjoy paying slightly less if a spate of buy or sell orders suddenly comes in. Unless there is a severe market breakdown, market orders are always filled.

■

A market order is also the simplest way to buy options and futures. Recall in the previous example that Shorty placed a market sell order and sold the option for $4, which was the bid price. If gold futures prices are quoted at $1,500.10 to sell and $1,500.00 to buy, then a market buy order is likely to get executed at $1,500.10.

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A Price-Contingent Order ■

Sometimes you may get a better price by waiting. You can place a limit order that must be executed at the stated or a better price or not executed at all. One must specify whether this is a day order or a good-until-canceled order. An unfilled day order automatically cancels at the day’s end, while a good-until-canceled (GTC) order stays open almost indefinitely in the specialist’s order book until it is closed. To eliminate stale orders, many brokers automatically cancel GTC orders after a significant time period has passed, for example, 60 days.

■

Suppose you place a limit buy order at $99.00. If the stock price decreases from $100.10 and your trade gets executed, then you obtained a $1.10 price improvement. However, you also run the risk that the stock may rally and the order may never be filled. Similarly for an option on YBM that is being quoted at a bid price of $4.00 and an ask price of $4.05, a trader hoping to benefit from market fluctuations may place a limit sell order at $4.50.

A Time-Contingent Order ■

A time-of-day order specifies execution at a particular time or interval of time. For example, you may request execution at 2:00 pm in the afternoon, or you may request execution between 2:30pm and 3:00pm.

Combinations of Related Orders

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■

Sometimes traders place orders in combination. For example, a spread order may instruct the broker to execute the order when the price difference between two securities reaches a certain value. A cancel order or a straight cancel order completely cancels a previous order. A cancel replace order replaces an existing order with a new one.

Our discussion also applies to options and futures trading, and we can similarly place such orders for bonds, the Treasury securities introduced in Chapter 2.

5.8

Summary

1. A call option gives the buyer (the “long”) the right but not the obligation to buy a specified quantity of a financial or real asset from the seller (the “short”) on or before a fixed future expiration date by paying an exercise price agreed today. A put gives the holder the right to sell. A put is similar to an insurance contract that pays in case of a decline in an asset’s value. The writer is paid a premium for selling such rights. 2. A European option can only be exercised at the maturity date, whereas an American option can be exercised at any time up to and including the expiration date. The names “American” and “European” do not depend on where the option trades. Most exchange-traded options in the US are of the American type. 3. A call holder’s payoff at expiration or if exercised early, its intrinsic value, is the stock price minus the strike price when the stock price is greater than the strike (call is in-the-money) and zero otherwise (call is out-of-the-money). A put’s

QUESTIONS AND PROBLEMS

intrinsic–exercise value is the strike price minus the stock price when the stock price is less than the strike and zero when the stock price is greater than or equal to the strike. Intrinsic value is only part of an option’s value. The rest is the time value, which takes into account that the option’s value may increase before it expires. 4. Payoffs to different option positions as a function of the stock price at expiration generate well-known patterns, which can be depicted in a payoff diagram. Profit diagrams adjust these diagrams for the option’s premium. 5. Traders in a hurry place market orders that are immediately executed. Traders who can wait in the hope of a better price strategically submit limit orders that may be contingent on price, time, and other relevant factors or submitted in combination with other orders. There is a trade-off: market orders may get traded at a bad price, while price improvement orders may never be filled. Contingent orders are possible for trading stocks, options, and futures.

5.9

Cases

Goldman, Sachs, and Co.: Nikkei Put Warrants—1989 (Harvard Business School

Case 292113-PDF-ENG). The case illustrates how investment banks can design, produce (hedge), and price put warrants on the Nikkei Stock Average. Nextel Partners: Put Option (Harvard Business School Case 207128-PDF-ENG).

The case examines issues surrounding Nextel’s shareholders’ vote to exercise a put option that requires the company’s largest shareholder, Sprint Nextel Corp., to purchase all the shares it does not already own. Copyright © 2018. World Scientific Publishing Company Pte. Limited. All rights reserved.

Pixonix Inc.—Addressing Currency Exposure (Richard Ivey School of Business

Foundation Case 908N13-PDF-ENG. Harvard Business Publishing). The case discusses the impact of exchange rate fluctuations on a firm’s cash flows and how derivatives can be used to manage this risk.

5.10

Questions and Problems

5.1. Explain the differences between an option’s delivery and expiration dates. 5.2. Explain the differences between European and American options. How do the

names relate to the geographical continents? 5.3. Explain the differences between call and put options. Are they just the exact

opposites of each other? 5.4. Explain the differences between long and short in the stock market. Explain

how one shorts a stock. 5.5. Explain the differences between long and short in the option market. What is

the difference between an option buyer and writer? 5.6. Explain the differences between an option’s intrinsic and time values. How do

these relate to the option’s price?

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5.7. Explain the differences between an option’s maturity and exercise date. When

does exercise take place for a European option? For an American option? 5.8. Explain and carefully describe the following four security positions, drawing

payoff diagrams wherever necessary to support your answer: a. short a forward contract with a delivery price of $100 b. short selling a stock at $100 c. going short on an option with a strike price of $100 5.9. Suppose that the current price of platinum is $400 per ounce. Suppose you

expect that in three months the price will increase to $425. You are worried, however, that there is a small chance that platinum may fall below $390 or even lower. What securities can you use to speculate on the price of platinum? 5.10. Why is an American option worth at least as much an otherwise similar

European option? Is there an exercise strategy that one can use to turn an American option into a European option? Explain. 5.11. You ask your broker for a price quote for (a fictional company) Sunstar Inc.’s

March 110 calls. She replies that these options are trading at $7.00. If you want to buy three contracts, what would be your total cost, including commission? Assume that the broker charges a commission equal to a flat fee of $17 plus $2 per contract. 5.12. “Because a call is the right to buy and a put is the right to sell, a long call

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position will be canceled out by a long put position.” Do you agree with this statement? Explain your answer. 5.13. The short position in a put option has the obligation to buy if the option is

exercised. Isn’t this counter to the notion that a short position indicates a “sell”? Explain your answer. 5.14. The following option prices are given for Sunstar Inc., whose stock price equals

$50.00: Strike Price

Call Price

Put Price

45

5.50

1.00

50

1.50

1.50

55

1.00

5.50

Compute intrinsic values for each of these options and identify whether they are in-the-money, at-the-money, or out-of-the-money. 5.15. Do you agree with the following statement: “An out-of-the-money option has

an intrinsic value of zero and vice versa”? Explain your answer.

QUESTIONS AND PROBLEMS

5.16. Compute the profit or loss on the maturity date for a short forward position

with a forward price of $303 and a spot price at maturity of $297. 5.17. Compute the profit or loss on the maturity date for a December 45 call for

which the buyer paid a premium of $3 and a spot price at maturity of $47. 5.18. Compute the profit or loss on the maturity date for a November 100 put for

which the seller received a premium of $7 and a spot price at maturity of $96. 5.19. Because options are zero-sum games, the writer’s payoffs are just the negatives

of the sellers. Demonstrate the following (by algebraic arguments or by using numbers): a. the call seller’s payoff at expiration is –max(0, S – K) = min(0, K – S) b. the put seller’s payoff is –max(0, K – S) = min(0, S – K) 5.20. What is the difference between a market buy order and a limit order? When

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would you use a market versus a limit order?

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6 Arbitrage and Trading 6.1 Introduction

Spread Trading

6.2 The Concept of Arbitrage

Index Arbitrage

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6.3 The No-Arbitrage Principle for Derivative Pricing The Law of One Price Nothing Comes from Nothing

EXTENSION 6.1 Stock Indexes EXTENSION 6.2 Index Arbitrage

6.6 Illegal Arbitrage Opportunities

6.4 Efficient Markets

6.7 Summary

6.5 In Pursuit of Arbitrage Opportunities

6.8 Cases

The Closed-End Fund Puzzle

6.9 Questions and Problems

THE CONCEPT OF ARBITRAGE

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6.1

Introduction

Once upon a time in days not too long ago, a venerable finance professor went with two of his graduate students to a McDonald’s restaurant for a working lunch. One student noticed that the Big Mac sandwiches were on a buy-one-get-one-free special. Bubbling with excitement, the student ran to the mentor to share the news. His jaws dropped as he approached, for the master himself was sitting with two Big Macs, discussing the selling of one to the second student at half price: “You also know about this?” “Yes,” the professor replied, “it’s an arbitrage.” Buying two for the price of one and selling the second free sandwich at half price would indeed be an arbitrage opportunity! Taking a cue from this embellished story, we define arbitrage (arb) as a chance to make riskless profits with no investment. The concept of arbitrage is invaluable for building pricing models, which are mathematical formulas that price derivatives in terms of related securities. Since the 1970s, financial markets have been generating innovative derivatives by attaching extra features to simple financial securities, by designing entirely new derivatives with complicated payoffs, or by combining several derivatives into a single composite security. For this last example, the composite security’s price must equal the sum of the prices of these simpler constituent derivatives. Otherwise, eagle-eyed traders can make arbitrage profits by buying the relatively cheaper of the two and selling the more expensive one. This observation leads one to recognize that a more general truth holds. In the absence of arbitrage, all portfolios or securities with identical future payoffs must have identical values today. This is the essence of the no-arbitrage principle that we discuss later in this chapter. Easy arbitrage opportunities are hard to find. Still we start with them because they form the foundations of our study and help us understand the more complex arbitrages that appear in today’s marketplace. The existence of arbitrage opportunities implies that the markets are somehow inefficient. We introduce the various notions of market efficiency and discuss what happens when they are violated. Next we talk about some pricing anomalies and the trading strategies that profit from them.

6.2

The Concept of Arbitrage

Financial markets exert a ruthless discipline on their players. No one advises a careless trader to modify bad quotes; instead, people grab such mispricings, and the misquoting trader sees his wealth disappear. Earlier we defined arbitrage as a chance to make riskless profits with no investments. This trivial description masks an activity that makes and takes millions of dollars every day. Eager traders pour over computer screens displaying prices from diverse markets and scan them for arbitrage opportunities. The act of arbitrage supports the livelihood of many. But gone are the days when one could sit in an armchair reading the financial press and find easy arbitrages. Today’s arbitrageurs work very hard for their money.

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Traders searching for arbitrage is rational and predictable behavior. Picking up a hundred dollar bill from the ground is normal behavior.1 It is hard, if not impossible, to model behavior if one oftentimes picks up the bill, sometimes doesn’t, and yet at other times picks up the bill but leaves a few bucks in its place for the person coming next! Therefore, for the subsequent theory, we assume that one never leaves money behind and relentlessly picks up even the tiniest of arbitrage opportunities. As we later see, this assumption has very powerful implications; indeed, it lies at the heart of the derivative pricing models presented in this book. There are many types of arbitrage. Some arbitrage operations are easy to understand and implement but hard to find in practice, whereas others are just the opposite. All such arbitrages can be decomposed into one of two types involving mispricings across space and mispricings across time. An example of mispricing across space is when the sum of the parts’ prices differ from the price of the whole. Example 6.1 discusses these two types of arbitrage opportunities.

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EXAMPLE 6.1: The Two Types of Arbitrage Opportunities ■

Arbitrage across space (see Figure 6.1). Suppose Your Beloved Machines Inc.’s (YBM) stock has an ask price of $100.00 on the Big Board and a bid price of $101.00 in Tokyo. If the brokerage cost is $0.10 per share, then one can buy stocks at a lower price in New York, simultaneously sell them at a higher price in Tokyo, and make 80 cents in arbitrage profits.

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Example (arbitrage across space). The sum of the parts’ prices differs from the price of the whole. Some people see arbitrage opportunities in conglomerates, which are huge companies with diverse lines of business. They claim that the share prices of conglomerates often fail to reflect the true value of the underlying assets and trade too low. Corporate raiders buy conglomerates, sell the constituent businesses or their assets, and more than recover their initial investments. “I saw the company and thought that the sum of the parts was worth more than what it was trading for,” observed a notorious corporate raider Asher Edelman as he targeted a French conglomerate.2 Investment bankers doing mergers and acquisitions believe the opposite. They believe that mergers create arbitrage opportunities where the sum of the parts is less than the whole. The idea that the same payoffs no matter how they are created, trade at the same price is known as the law of one price.

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Arbitrage across time (see Figure 6.1). Free lottery tickets are an example of arbitrage across time. Result 3.1 of Chapter 3, which states that a stock’s price falls exactly by the amount of the dividend on the ex-dividend date, if violated, also represents an example of an arbitrage across time.

1

In an often-told tale, two finance professors are walking together, and they see a $100 bill lying on the ground. When one ponders whether he should pick it up, the other replies not to bother because the bill is not really there: “As markets are efficient, someone must have already picked it up before.” This story is usually told to clarify the “ideal” notion of an efficient market, while teasing finance academics.

THE CONCEPT OF ARBITRAGE

FIGURE 6.1: The Two Types of Arbitrage Opportunities Security trades at some price Buy cheap, sell expensive; make net profit after paying brokerage costs Same security trades elsewhere at a lower price

i)

ii)

Now

Later

Zero net investment

Nonnegative payoff always and strictly positive payoff with positive probability

Positive cash flows

Nonnegative payoff always

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2

Corporate raider Asher B. Edelman once taught a business course at Columbia University, in which he offered $100,000 to any student who would find a takeover target for him. However, the university banned the offer. In the late 1990s, he targeted Societe du Louvre, the French holding company to which the above comment applies (see “Champagne Taste: An American Raider in Paris Challenges an Old World Dynasty—Edelman Presses Taittingers to Choose Family Values or Shareholder Value—Free to Paint a Hotel Pink,” Wall Street Journal, November 11, 1998).

Are such arbitrage opportunities likely to exist in today’s financial market, where technological advances have lowered brokerage costs and made it possible to move millions of dollars in a matter of seconds? Most economists believe that the answer to this question is no—and if they exist, it is especially hard to find them in wellfunctioning and efficient markets. Perhaps they can be found in some emerging markets, but they fade away quickly as traders start grabbing them. Sometimes what may seem like arbitrage opportunities in a perfect world (where market imperfections or frictions like transaction costs, bid/ask prices, taxes, and restrictions on asset divisibility are assumed away) cease to be so when market imperfections are included. In actual markets, market frictions are relevant and should always be taken into account when considering arbitrage opportunities. Let’s discuss a few real-life examples of arbitrages.

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EXAMPLE 6.2: Real-Life Arbitrages ■

Sports arbitrage. Not too long ago the soccer World Cup finals had two powerhouse teams, Brazil and Germany, playing. Uncle Sanjoy relishes the story of how he visited a Wall Street firm and entered into two bets with acquaintances. First, he agreed to pay one person $10,000 if Brazil won but receive $5,000 otherwise. Then, uncle slyly adds that he visited another office, and bet with a second person to receive $11,000 if Brazil won but pay $4,000 otherwise. No matter which team won, uncle locked in $1,000. Although gambling exasperates his wife, uncle quietly commented that he had locked in arbitrage profits by exploiting inefficiencies in the system and that he truly enjoyed watching the game!

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Airline miles arbitrage. Airlines put all kinds of restrictions on tickets, serve scant morsels of food, manhandle baggage, make passengers endure long delays, and cancel flights. Those who have suffered in the sky at the hands of airlines will be amused by a story in the Wall Street Journal, “How Savvy Fliers Make the Most of Their Miles” (December 16, 2008), which related how some travelers got the best out of their frequent flier miles. Normally a ticket obtained by redeeming frequent flier miles would yield the value of approximately one cent per mile. Some folks figured out how to transform this once cent per mile into more. One gentleman paid about 1.3 cents per mile to “friends” through the Internet and then used the miles for business class tickets between China and the United States. “I call it airline miles arbitrage,” said the savvy traveler, who created business class tickets for coach ticket prices and in the process, earned about six to nine cents per mile.

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Uncle Sam’s generosity. The idea of arbitraging Uncle Sam (the US government) appeals to most people, but few succeed in doing so. However, an article in the Wall Street Journal, “Miles for Nothing: How the Government Helped Frequent Fliers Make a Mint—Free Shipping of Coins, Put on Credit Cards, Funds Trip to Tahiti; ‘Mr. Pickles’ Cleans Up” (December 7, 2009), reported a story where this happened. What happened? Because a paper dollar lasts only about twenty-one months in circulation, dollar coins save a country money as they can last for thirty years or more. To promote their use, the US Mint offered to sell presidential and Native American $1 coins at face value, ship them for free, and buyers could charge the purchase to their credit cards without handling fees. However, wellintentioned plans sometimes have unintended consequences. Arbitrageurs bought large quantities of coins from the mint, paid for them with credit cards that offered frequent flier miles for free air travel (typically, one cent for each dollar charged in purchases), and immediately deposited the coins in the bank to pay for the credit card coin purchase. The story states that one software consultant ordered $15,000 worth of coins and even had the delivery person deposit the coins directly in his car’s trunk so that he could take them to the bank. Discovering such activities, US Mint officials sent letters to prospective buyers seeking the reasons for their purchase and denied further access to those who did not respond.

THE NO-ARBITRAGE PRINCIPLE FOR DERIVATIVE PRICING

6.3

The No-Arbitrage Principle for Derivative Pricing

Modern finance theories use the idea that arbitrage opportunities do not exist in well-functioning markets to derive many important results. This was pioneered by the Miller and Modigliani (M&M) propositions in corporate finance, which state that if market imperfections like taxes are assumed away, then the choice of debt or dividend policy does not affect the value of a firm. (Taxes are imperfections! Who says finance jargon is boring?) In our case, the absence of arbitrage opportunities lies at the heart of derivative pricing models, including the celebrated Black–Scholes– Merton and the Heath–Jarrow–Morton models included in this text. This section explains how.

The Law of One Price

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Suppose one wants to price a market-traded derivative, which we label as “portfolio A.” Assume that there are no cash flows on this traded derivative before maturity so that there are only two dates of concern—today and when the derivative matures. Now let us create portfolio B, a collection of different traded securities with known prices and no intermediate cash flows, to match portfolio A’s payoffs on the maturity date (see Figure 6.2). The values of these two portfolios are linked by a no arbitrage principle: to prevent arbitrage, two portfolios with identical future payoffs must have the same value today.3 Using this principle, the cost of creating portfolio B is the “fair” or “arbitrage-free” value of the traded derivative portfolio A.4 This is known as the law of one price.

Nothing Comes from Nothing Continuing with the previous discussion, if we subtract portfolio A from portfolio B, then we get the second representation, which will be one of our main methods for proving results (see Figure 6.2). By construction, the value of this new portfolio is zero for sure on the maturity date. To prevent arbitrage profits, it must also have zero value today. The derivative’s price can be solved from this condition. This insight can be paraphrased as nothing comes from nothing.5 When identical cash flow securities are subtracted, yielding zero at a future date, the result is emptiness, nothing, nada, zero now! These are the two standard techniques used for pricing derivatives from Chapter 11 onward in this book. 3

Suppose this principle does not hold. If portfolio A is cheaper than portfolio B today, “buy low, sell high”: buy A and sell B. You will have a zero cash flow in the future because the cash inflow will cancel the outflow, and the price difference you capture today becomes arbitrage profit. If B is cheaper than A, just reverse the trades to capture the arbitrage profit. 4

The same argument works when the price of the derivative is known but there is an unknown variable in the payoff. For example, a forward contract has a zero value on the starting date, but the unknown forward price is in the payoff. The solution to this equation is the arbitrage-free forward price. 5 “Nothing comes from nothing” (or ex nihilo nihil fit in Latin) is a philosophical expression of a thesis first advanced by the Greek philosopher Parmenides of Elea, who lived in the fifth century bc.

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FIGURE 6.2: Pricing Derivatives The law of one price Today

Later date

Portfolio A

Must be equal; No intermediate cash flows otherwise arbitrage

Portfolios have the same final value

Portfolio B

Nothing comes from nothing Now

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Portfolio A – (Portfolio B)

Later date No intermediate cash flows

Zero value (for sure)

Must be zero; otherwise arbitrage

As we have often observed, arbitrage opportunities are likely to be absent in wellfunctioning and efficient markets—but what do we really mean by an “efficient market”? Well, read on.

6.4

Efficient Markets

When a finance academic states that “markets are efficient,” she means an informationally efficient market: a market where asset prices quickly absorb and “fully” reflect all relevant information. But what relevant information is this referring to? The efficient markets hypothesis (EMH) is presented with respect to three different information sets: 1. Weak-form efficiency asserts that stock prices reflect all relevant information that can be gathered by examining current and past prices. If this is true, then technical analysis (which involves looking at price patterns to predict whether a stock’s price will go up or down) will not generate returns in excess of the

IN PURSUIT OF ARBITRAGE OPPORTUNITIES

risk involved. In addition, and more relevant to us, if the market is weak-form efficient, then there are no arbitrage opportunities. Why? Because finding arbitrage opportunities only depends on the information contained in current and past prices. It is this connection that relates an efficient market to the arbitrage-free pricing of derivatives. 2. Semistrong-form efficiency asserts that stock prices reflect not only historical price information but also all publicly available information relevant to those particular stocks. If this holds, then fundamental analysts (fundamental analysis involves reading accounting and financial information about a company to determine whether a share price is overvalued or undervalued) would be out of business.

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3. Strong-form efficiency asserts that stock prices reflect all relevant information, both private and public, that may be known to any market participant. If this is true, then even insiders (who have privileged, yet to be made public information about the company) will not make trading profits in excess of the risk involved. There is a never-ending debate on whether markets are efficient with respect to these three different information sets. The preponderance of evidence tends to accept weak-form efficiency, and strongly reject strong-form efficiency, especially given the numerous insider-trading lawsuit convictions historically obtained by the Securities and Exchange Commission (SEC). The evidence with respect to semistrong-form efficiency is mixed. The assumption of no arbitrage underlying the models developed in this book is very robust. Indeed, if markets are weak-form efficient, then no arbitrage opportunities exist. Hence the weakest form of market efficiency justifies the assumption of no arbitrage. But don’t get confused. Assuming there are no arbitrage opportunities in subsequent modeling does not imply that we are assuming that the market is weak-form efficient. As just discussed, the assumption of no arbitrage is a much weaker notion than even a weak-form efficient market. Hence it is more likely to be satisfied in actual markets, and thus it provides a more robust model than market efficiency does.

6.5

In Pursuit of Arbitrage Opportunities

Well-resourced, astute players trade for the purpose of exploiting arbitrage opportunities. As the next section illustrates, academics have uncovered phenomena that seem to suggest the existence of such arbitrages.

The Closed-End Fund Puzzle Closed-end fund share prices are often claimed to represent arbitrage opportunities. To understand this claim, we must first understand closed-end funds, and to understand closed-end funds, we need to first understand investment companies. An investment company sells shares to investors, invests the funds received in various securities according to a stated objective, and passes the profits and losses back to the investors after deducting expenses and fees. Federal securities law classifies investment companies into three basic types: mutual funds (legally known as open-end

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companies), closed-end funds (legally known as closed-end companies), and UITs (legally known as unit investment trusts). Mutual funds, like those offered by Fidelity and Vanguard, are open ended and take new investments at any time. A shareholder can redeem his shares directly from the fund whenever he desires. Closed-end funds issue shares at their creation but none thereafter. Just like regular stocks, the shares of closed-end funds trade on organized exchanges. Closed-end funds are not redeemed by the fund for cash. To close out an investment, an investor must sell her shares on the secondary market. Finally, unit investment trust shares can be redeemed like mutual funds; moreover, they have a termination date when the fund’s assets are sold and the proceeds are paid out to investors. Redemptions in mutual funds and UITs pay the company’s net asset value per share. Net asset value (NAV) is the value of the company’s total assets minus its total liabilities, often expressed on a per share basis. For example, if an investment company has securities and other assets worth $110 million and has liabilities of $5 million, its NAV would be $105 million. If the investment company has issued 1 million shares, then $105 would be its NAV per share. NAV changes daily due to market fluctuations. Mutual funds and UITs are usually required to compute their NAV at least once per business day (which they usually do after the market closes). Closed-end funds, however, are exempted from this requirement. For reasons that are doubted and debated, the shares in most closed-end funds trade at a discount from their asset value. This is known as the closed-end fund puzzle. These discounts represent potential arbitrage opportunities. To see this, suppose that a closed end fund is worth $100 million in the market, but the underlying securities have a net asset value of $105 million. What can one do? One can buy all the shares, open up the fund, and sell the underlying securities to make $5 million in arbitrage profits. However, not all closed-end funds trade at a discount. Some trade at premiums over their underlying asset values, like the legendary Warren Buffett’s Berkshire Hathaway Company at the end of the twentieth century. Why? Again, a puzzle.

Spread Trading At the heart of many arbitrage strategies is spread trading. Spread trading involves buying and selling one or more similar securities when their price differences are large, with the aim of generating arbitrage profits as their price differences later converge to some target value. For example, suppose that Your Beloved Machines Inc. (YBM) has two classes of shares with different voting rights. You find that class A shares are trading at $100 and class B shares are trading at $85. This is a spread of 100 – 85 = $15. However, you’ve discovered that the spread has historically been $10, and you expect it to return to that level. To take advantage of this price discrepancy, you can buy the underpriced class B shares and sell the overpriced class A shares. If your bet is correct and the spread returns to $10, you can reverse the trades and capture $5 in profits. Notice that you are betting on the movement of a spread and not on the direction of the stock prices

IN PURSUIT OF ARBITRAGE OPPORTUNITIES

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themselves. Spread trading is one of the most common arbitrage trading strategies used in financial markets. The next section discusses one such widely used strategy, called index arbitrage.

Index Arbitrage A stock index is an average of stock prices that are selected by some predetermined criteria. For example, the Dow Jones Industrial Average (also called the DJIA, Dow 30, the Dow Industrials, the Dow Jones, or simply the Dow) and the Standard and Poor’s 500 Index (S&P 500) are stock indexes created by using a price-weighted average of thirty major US stocks and the market value–weighted average of five hundred major US companies, respectively (see Extension 6.1 for a discussion of indexes). Sometimes the value of a stock market index gets disjointed from the arbitrage-free price of a futures contract written on the index. In such situations, the potential for index arbitrage exists. Suppose the traded futures contract on an index is overvalued. The arbitrage is to short the traded futures contract and go long a synthetic futures contract constructed to have the same cash flows as the traded futures contract. The synthetic futures contract is constructed using the portfolio of the traded stocks underlying the index. This strategy will lock in the price difference at time 0 and have zero cash flows thereafter. The arbitrage is a violation of the nothing comes from nothing principle. Extension 6.2 illustrates index arbitrage via a simple example.

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EXTENSION 6.1: Stock Indexes There are two basic types of stock indexes: price-weighted and value-weighted indexes. We illustrate both using examples.

EXT. 6.1 EX. 1: A Price-Weighted Index (PWI) ■

Holding one share of each stock, summing the share prices and dividing by the total number of shares creates a price-weighted index. Consider the hypothetical stock price data in Ext. 6.1, Table 1. Here YBM’s share price is $100 today and $110 tomorrow. BUG’s price is $50 today and $45 tomorrow. The equal average PWI is equal to Sum of stock prices / Number of stocks = (100 + 50) /2 = 75 today and (110 + 45) /2 = 77.5 tomorrow

■

While one stock went up and the second went down, the PWI index registered a 2.5 point gain or (Tomorrow’s index value – Today’s index value) / Today’s index value = (77.5 – 75) / 75 = 0.0333 or a 3.33 percent increase

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This is called a price-weighted average because when viewed as a portfolio, each stock’s return is weighted by a percentage proportionate to the price of the stock relative to the value of the portfolio.

The DJIA is computed by creating a price-weighted average of thirty high-quality blue chip stocks from diverse industries. The component stocks are selected by the editors of the Wall Street Journal. The membership list is periodically modified so that the index continues to accurately reflect the general US stock market. Such revisions are imperative because companies go through mergers, acquisitions, and bankruptcy.

EXT. 6.1 TABLE 6.1: Stock Price Data Stock

Today’s Price

Tomorrow’s Price

Shares Outstanding (millions)

Today’s Market Value (millions)

Tomorrow’s Market Value (millions)

YBM

$100

$110

50

$5,000

$5,500

BUG

$50

$45

20

$1,000

$900

$6,000

$6,400

Total

EXT. 6.1 EX. 2: A Value-Weighted Index (VWI)

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■

Multiplying the stock price by the shares outstanding (which gives the stock’s total market capitalization) and then summing across all stocks creates a value-weighted index. Using the data from Ext. 6.1 Tab. 1, we get a VWI equal to Sum of stock price times shares = (100 × 50 + 50 × 20) = 6, 000 today and (110 × 50 + 45 × 20) = 6, 400 tomorrow The weights (percentage holdings of each stock) in the index are for YBM = $5, 000/$6, 000 = 0.8333 for BUG = $1, 000/$6, 000 = 0.1667

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The percentage change in the VWI is (Tomorrow’s index value – Today’s index value) /Today’s index value = (6, 400 – 6, 000) /6, 000 = 0.0667 or a 6.67 percent increase

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In practice, the index is often normalized to make its value equal to some arbitrary number at a given date. This is for convenience in reporting the index. For example, the VWI index’s value could be normalized to be 100 today by dividing by 60. The number 60 must also divide the index’s value at all future dates. In this case, the index value tomorrow would be 6,400/60 = 106.67.

IN PURSUIT OF ARBITRAGE OPPORTUNITIES

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This is called a value-weighted index because when viewed as a portfolio, each stock’s return is weighted by its market value relative to the total value of the stock market. The S&P 500 is a market-value-weighted index of five hundred large-sized US companies. A value-weighted index is sometimes called a market-cap weighted index.

EXTENSION 6.2: Index Arbitrage This extension explains index arbitrage. To simplify the presentation, the discussion replaces futures with forward contracts. For the purposes of this demonstration, this difference is not important. The differences between forward and futures contracts, which were discussed in Chapter 4, will be further clarified in subsequent chapters. Index arbitrage generates riskless profits by putting the no-arbitrage principle to work. We illustrate it with a simple example.

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EXT. 6.2 EX. 1: The No-Arbitrage Principle ■

Let a PWI be created by computing an average of the stock prices of YBM and BUG worth $100 and $50 today, respectively. This gives an index value of $75 today. - Suppose that a dealer quotes a forward price of $80 on a forward contract on PWI that matures in one year. - Let the simple interest rate be 6 percent per year. And suppose that neither stock will pay any dividends over the next year.

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Next, suppose that a trader believes the forward price is too high. Intuitively, she wants to sell the traded forward contract and create a synthetic forward contract by buying the PWI index portfolio, borrowing the funds to do so, and holding the portfolio until the traded forward contract matures. She performs the following steps: - She sells two forward contracts to the dealer, whereby she locks in a selling price of $80 per unit of the index: $160 in all. - Simultaneously, she buys one share of each stock for a total cost of $150 and finances this purchase by borrowing. Notice that she has a zero cash flow today because the traded forward requires no cash flows on its initiation date. - One year later, she gives two units of the index to the dealer who sold her the forward. This costs nothing because she made allowance for this by purchasing the stocks earlier. She gets $160 from him in return. - But she has to repay the loan with interest. This entails an outflow that equals Today’s index purchase price × (1 + interest cost) = 150 × 1.06 = $159 - She pays out $159 but receives $160, and makes $1 in arbitrage profits. She makes this profit no matter where the index moves a year from today. We make extensive use of this technique to develop cost-of-carry models in Chapters 11 and 12.

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6.6

Illegal Arbitrage Opportunities

Arbitrage is a broad concept. You can bring under its umbrella dubious, socially unfavorable, and illegal trading activities. Medical students first learn how a healthy body functions before they study abnormalities and diseases. We follow a similar approach. Previously we introduced legal arbitrage opportunities that facilitate market efficiency. Now we shift gears and study nefarious arbitrage opportunities. These illegal arbitrage opportunities occur in malfunctioning securities markets and remain a major problem that regularly appears in the national, local, and financial press. Trading abuses and manipulations are forms of illegal arbitrage opportunities that we now discuss. One such abuse is front running, which is trading based on an impending transaction by another person. For example, a broker may buy on his own account in front of his customer’s buy order. Then, he makes unscrupulous gains by selling the shares he bought to the customer at a higher price when executing the customer’s buy order. A chronic problem facing free markets is that individuals often try to manipulate prices to their advantage. Manipulation has long been a concern in the securities market. For example, in its report on the [Securities] Exchange Act of 1934, the Senate Committee wrote, “The purpose of the act is... to purge the securities exchange of these practices which have prevented them from fulfilling their primary function of furnishing open markets for securities where supply and demand may fully meet at prices uninfluenced by manipulation or control.”6 The Commodity Exchange Act states that it is illegal to manipulate or attempt to manipulate a commodity’s future delivery price, yet proving manipulation in a court of law is not easy. Next we discuss some well-known examples of market manipulation. A bear raid is a market manipulation strategy generating arbitrage profits that was widely prevalent in US stock markets in the nineteenth and early twentieth centuries. Remember from Chapter 2 how short selling works—a short seller borrows shares, accepts the obligation to pay dividends to the lender, and agrees to return the security to the lender at a future date. Traders organizing a bear raid take large short positions in a company’s stock, spread unfavorable rumors that depress the stock further, and buy back those shares when other shareholders panic and unload their stock holdings at depressed prices. Short sellers can get severely burned if they get caught in a market corner and a short squeeze, which was also a pervasive problem in the early days of stock markets. The next example shows how a market corner and short squeeze works.

6

Report of the Committee on Banking and Currency, “Stock Exchange Practices,” U.S. Senate Report No. 1455, 73d Cong., 2d Sess., 1934 (the “Fletcher Report”). The quote appears in Chapter I “Securities Exchange Practices,” Section 11 “The government of the exchanges,” Subsection (b) “Necessity for regulation under the Securities Exchange Act of 1934.”

ILLEGAL ARBITRAGE OPPORTUNITIES

EXAMPLE 6.3: A Market Corner and Short Squeeze ■

Suppose a company has ten thousand outstanding shares, which sell for $50 in the market. This is the deliverable supply, of which Alexander owns three thousand shares, Bruce owns three thousand shares, and Caesar owns four thousand shares. The shares are held in street name with their common broker, Mr. Brokerman, so that short sellers can borrow (see Figure 6.3).

FIGURE 6.3: Example of a Market Corner and Short Squeeze

Shorty borrows shares from Alexander

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Alexander 3,000 shares

Caesar 4,000 shares

Bruce 3,000 shares

Shorty Short 3,000 shares

Sells shares to Ruth Ruth owns 13,000 shares Short sells to Ruth

Ruth controls 13,000 shares, though the deliverable supply is 10,000 shares. Ruth has cornered the market. She will squeeze Shorty by charging him a high price for 3,000 shares that he has to buy and return to Alexander.

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Shorty is bearish on the company and expects the share price to decline from $50 to $40. He short sells three thousand shares, which his broker, Mr. Brokerman, borrows from Alexander. They are sold at $50 per share to Ruth, who happens to be a “ruthless manipulator.”

■

Ruth now buys all the shares from Alexander, Bruce, and Caesar. She now has a monopoly position and has bought three thousand more shares than the deliverable supply of ten thousand shares, making it thirteen thousand in all. Ruth has cornered the market.

■

At the time of settlement, Alexander does not have shares to give to Ruth. He asks Mr. Brokerman to return his three thousand shares. Since Ruth owns them all, nobody is willing to lend shares. Normally, Shorty can keep his short position open as long as he likes. As scarcity is preventing Alexander’s trade from getting settled, Brokerman informs Shorty that he has to return the borrowed shares at once.

■

Shorty has nowhere to go but Ruth to buy shares. This is a case of congestion: Shorty, trying to cover his short position, finds that there is an inadequate supply of shares to borrow, and Ruth is unwilling to sell her shares, except at sharply higher prices. In other words, Ruth is squeezing the short. She can easily bankrupt Shorty.

This example easily extends to futures market. First the trader goes long the futures contract in excess of the immediately deliverable supply. Then she keeps her long position open, eventually acquires all the deliverable supply, and ends up in a monopoly position. After she has cornered the market, a short squeeze develops. She demands delivery, but the short cannot find any supply to cover their positions, except from her—and she extracts her price. These undesirable situations are in violation of the Commodity Exchange Act statute. To stop short squeezes, the CFTC can suspend trading and force settlement at a “fair” price set by the exchange issuing the contract. This happened, for example, with the Hunt brothers’ holdings of silver futures in 1979 and 1980. Contract provisions are also designed to increase the deliverable supply, which minimizes the likelihood of manipulation, by allowing variation in the quality of the asset delivered (with appropriate price adjustments) and a longer delivery period. Still, manipulations happen. We discuss additional futures market manipulations in Chapter 10.

6.7

Summary

1. Arbitrage is a chance to make riskless profits with no investment. Arbitrage profits can be generated in several ways—across time and across space. An example of an arbitrage across space is when the sum of the parts’ prices differ from the price of the whole. 2. The no-arbitrage pricing principle involves skillfully creating a portfolio of assets that replicates the payoff to a traded derivative and then arguing that the cost of this portfolio is the “fair” or “arbitrage-free” value of the traded derivative. This is the standard technique used for pricing derivatives.

QUESTIONS AND PROBLEMS

3. Arbitrage profits do not exist in an efficient market where market prices reflect all past and current price information. However, markets aren’t always efficient, and sophisticated traders use advanced tools and techniques to make arbitrage profits. 4. A chronic difficulty with free markets is that traders try to manipulate prices to their advantage. Market manipulation is an example of an illegal arbitrage opportunity. 5. A market corner and short squeeze is a common manipulation technique that has a checkered past in futures markets. It involves a trader going long in the futures market and squeezing the shorts subsequently in the cash market, when they scramble to cover their short positions.

6.8

Cases

Arbitrage in the Government Bond Market? (Harvard Business School Case 29-

3093-PDF-ENG). The case examines a pricing anomaly in the large and liquid Treasury bond market, where the prices of callable Treasury bonds seem to be inconsistent with the prices of noncallable Treasuries and an arbitrage opportunity appears to exist. RJR Nabisco Holdings Capital Corp.—1991 (Harvard Business School Case 29-

2129-PDF-ENG). The case explores a large discrepancy in the prices of two nearly identical bonds issued in conjunction with a major leveraged buyout and considers how to capture arbitrage profits from the temporary anomaly.

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Nikkei 225 Reconstitution (HBS Premier Case Collection Case 207109-PDFENG).

The case considers how an institutional trader who may receive several billion dollars of customer orders in connection with a redefinition of the Nikkei 225 index can provide liquidity for the event and pick up arbitrage profits.

6.9

Questions and Problems

6.1. What is an arbitrage opportunity across space? Give an example. 6.2. What is an arbitrage opportunity across time? Give an example. 6.3. What is the law of one price? 6.4. Explain the pricing principle “nothing comes from nothing.” 6.5. Suppose a two-year Treasury note is trading at its par value $1,000. You

examine the cash flows, and if you sell them individually in the market, you get $47.85 for the six-month coupon, $45.79 for the one-year coupon, $43.81 for the one-and-a-half-year coupon, $41.93 for the two-year coupon, and $838.56 for the principal. a. Are these prices correct? b. If not, show how you can capture arbitrage profit in this case.

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6.6. Suppose a two-year zero-coupon bond has a price of $0.90 and a three-year

zero has a price of $0.85. A bank allows you to borrow or lend at 4 percent, compounded once a year. Show two ways that you can make arbitrage profits from these prices. 6.7. You can trade Boring Unreliable Gadget Inc.’s stock for $77 per share in the

United States and for €50 in Europe. Assume a brokerage commission of $0.10 per share in the United States and €0.10 in Europe. A foreign exchange dealer quotes a bid price of $1.5000 for each euro and offers them for $1.5010. a. Are these prices correct? b. If not, show how you can capture arbitrage profit by trading BUG stock. 6.8. What is an efficient market, and what does it mean for a market to be weak-

form, semistrong-form, and strong-form efficient? 6.9. If the market is weak-form efficient, do arbitrage opportunities exist? Explain

your answer. Do you think arbitrage opportunities exist? Explain your answer. 6.10. What is a closed-end fund, and what is the “closed-end fund puzzle”? 6.11. Suppose that Boring Unreliable Gadget Inc. has two classes of shares with

different voting rights. You find that class A and class B shares are trading at $49 and $37, respectively. However, historically, the spread has been $15, and you expect the price difference to reach that level. a. Explain how you would set up a spread trade and how much profit you

expect to make once the prices correct themselves.

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b. Would the preceding strategy work if class A stock goes up to $75 per share? 6.12. What is a stock index? Describe the differences between the Dow Jones

Industrial Average and the S&P 500 stock price indexes. 6.13. What is algorithmic trading, and how can it be used to make profits? How does

this relate to program trading? 6.14. What is front running? Explain why an investor should be concerned if his

stockbroker front runs his trades. 6.15. What is a short squeeze, and under what circumstances does it occur? 6.16. What is a bear raid? How does a bear raid relate to trading on inside

information?

7

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Financial Engineering and Swaps 7.1 Introduction

Interest Rate Swaps

7.2 The Build and Break Approach

Forex Swaps

7.3 Financial Engineering

Currency Swaps

Cash Flows versus Asset Values

EXTENSION 7.1 Valuing Fixed-for-Fixed Currency Swaps

Examples

Commodity and Equity Swaps

7.4 An Introduction to Swaps 7.5 Applications and Uses of Swaps 7.6 Types of Swaps

Credit Default Swaps

7.7 Summary 7.8 Cases 7.9 Questions and Problems

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7.1

Introduction

In January 1993, the French chemical and pharmaceutical giant Rhône-Poulenc offered its employees an opportunity to buy discounted shares as part of its forthcoming privatization plan. With an eye toward “wide employee participation,” the scheme involved a dizzying array of features, including minimum investment incentives, price discounts, free shares, interest-free loans, tax benefits, an option to opt in or out of dividends, and an installment plan to pay for the shares over time. Despite the best intentions of this intricate scheme, the plan badly flopped. Then came Bankers Trust, an American bank famous for its derivatives and trading prowess (and infamous for widely publicized losses incurred by its clients Procter & Gamble and Gibson Greetings). Bankers Trust streamlined the terms of the Rhône-Poulenc plan and also added a critical safety feature—the investment would earn a guaranteed minimum return but surrender some gains if the stock price increased. The modified plan was a success. How could they guarantee a minimum return? Armed with your derivatives knowledge, you can now understand what happened. The plan protected RhônePoulenc’s employees from the downside (by buying a put option), and it financed this plan by removing some of the upside (by selling a call option). This is an example of financial engineering. Financial engineering studies how firms design derivatives to solve practical problems and exploit economic opportunities. This vast subject is taught at varying depths in many universities around the world. After presenting some simple examples, we introduce swaps, which were among the first uses of financial engineering. We briefly discuss applications and uses of swaps and illustrate the basic workings of interest rate swaps, forex swaps, currency swaps (including a valuation formula), commodity swaps, equity swaps, and the highly useful but much maligned credit default swaps. Chapter 22 discusses swap markets and interest rate swaps again in greater detail.

7.2

The Build and Break Approach

Sometimes children snap together plastic blocks to build beautiful toys. At other times, they rip apart their creations into simpler structures. We adopt a similar build and break approach. Sometimes we combine derivatives and financial securities to create new derivatives tailored to meet investment needs, and at other times, we break down a complex derivative into its simpler components. This is important for two reasons. The first reason is that it provides an approach for pricing derivatives—break a derivative into simpler parts, price the simpler parts, and sum to get the original derivative’s price. Arbitrage, like adhesive, binds the parts’ prices together to equal the price of the whole and makes sure that the technique works. For example, you can price a convertible bond as the sum of a regular bond and an option to convert the bond into a fixed number of stocks. An extendible bond may be valued as the price of

FINANCIAL ENGINEERING

a straight bond plus an option to extend the bond’s life. So when a rocket scientist tries to sell you a complex derivative at an exorbitant price, you can value it by this method and prevent yourself from being gouged. The second reason is that it enables the use of derivatives to modify a portfolio’s return to meet various investment objectives. For example, suppose a company is considering issuing fixed rate debt when interest rates are high to finance an investment project. The company is worried about getting stuck with an expensive debt issue if interest rates go down. To avoid this problem, the company can issue callable bonds. In the case that interest rates decline, the company can exercise the call provision in the callable bonds and buy back the bonds at a predetermined price. Where does the company get the funds to buy back the bonds? It issues new fixed rate bonds at the lower interest rate.

7.3

Financial Engineering

Toward the end of the 20th century, financial engineering emerged as a vibrant, new subfield of finance and operations research.1 Financial engineering applies engineering methods to finance. At the simplest level, financial engineering and risk management puts our build and break approach to work. The best way to understand this approach is through some examples of financial engineering in practice.

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Cash Flows versus Asset Values To a beginning student, when forming and liquidating portfolios, the signs of the cash flows and asset values may be confusing. To preempt such confusion, we make the following observations. When constructing a portfolio, a positive value means that you are purchasing an asset, hence there is a negative cash flow. For example, if you are buying Your Beloved Machine stock, it shows up as a positive value in the portfolio, but you spend $100, a negative cash flow, in acquiring it. If the value is negative, you are creating a liability, and there is a positive cash flow. For example, borrowing $90 gives a positive cash inflow of $90. The $90 is a liability because you must pay it back in the future. Note that when a portfolio is constructed, the signs of the cash flows and asset values are opposite. At liquidation, this relationship changes. When liquidating a positive value position, you are selling an asset; hence there is a positive cash flow. For example, YBM has a positive value in your portfolio, and selling it for $120 will generate a positive cash flow. If the value is negative before liquidation, then the cash flow is negative because you are closing a liability. For example, your $90 loan, being a liability, has a negative value in the portfolio. Repaying it on the liquidation date, say, for $94, incurs a negative cash flow. Note that at liquidation, the signs of the cash

1

Cornell University offered the first financial engineering degrees. It states on the Cornell University website (www.orie.cornell.edu/orie/fineng/index.cfm) that “Robert Jarrow and David Heath advised students for several years before formalizing the program in 1995, making Cornell one of the very first universities to have a graduate program in Financial Engineering, and arguably the oldest such program in the world.”

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flows and asset values are the same. We will often call both the cash flows and values at liquidation the payoffs. As they have the same sign, no confusion should result. When proving results, for some arguments we will consider cash flows and for others asset values. As long as you keep these relationships in mind, there should be no confusion. The key is to “follow the money!”

Examples Example 7.1 considers a commodity derivative—a bond whose payoff is indexed to gold prices.

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EXAMPLE 7.1: A Bond Indexed to Gold Prices ■

Goldmines Inc. (fictitious name) is a gold mining company. It sells a highly standardized product, pure gold, in the world market. It needs money for exploring, for setting up a mine, for running a mine, and for refining operations. The company’s fortune ebbs and flows with the price of gold.

■

The company decides to raise cash by selling bonds. This is attractive because interest payments are tax deductible. But debt makes a company vulnerable to financial distress—a debt-free company, by definition, can never go bankrupt!

■

An investment banking firm designs a bond with the following payoff on the maturity date T: - A payment of $1,000 - An additional amount tied to gold’s price per ounce, S(T), which is 0 if S (T) < $950 $10 [S (T) − 950] if $950 ≤ S (T)

■

This payoff is given in Table 7.1 and graphed in Figure 7.1. You can generate the first payment synthetically by investing in zero-coupon bonds. The second payoff looks similar to a long call option position: buying ten calls with a strike price of $950 does the trick. Consequently, the cash flows from the traded bond may be obtained synthetically by forming a portfolio consisting of (1) a zero-coupon bond worth 1,000B, where B is today’s price of a zero-coupon bond that pays $1 at maturity, and (2) buying ten European calls, each with a price of c and a strike price of $950 (see Table 7.2).

■

The no-arbitrage principle assures that the replicating portfolio must have the same value as the traded bond. Therefore the bond’s value is (1,000B + 10c). In later chapters, you will learn how to price this call option with the binomial or the Black–Scholes–Merton model.

■

Why sell a security like this? First, it will raise more cash up front. Second, it hedges some of the output price risk that Goldmines Inc. faces. If gold prices are high at maturity, it shares some of its profits—that’s not too bad. And if gold prices decline, only the principal is repaid.

FINANCIAL ENGINEERING

TABLE 7.1: Payoffs for a Traded Bond Indexed to Gold Bond Payoff

Time T (Maturity Date) Cash Flow S(T) < 950

950 ≤ S(T)

1,000

1,000

0 Net cash flow

10[S(T) – 950]

1,000

1,000 + 10[S(T) – 950]

FIGURE 7.1: Payoff Diagram for a Bond Indexed to Gold Payoff Slope = 10

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1,000

0 950

S(T), spot price of gold at maturity

TABLE 7.2: Cash Flows for the Synthetic Bond Indexed to Gold Portfolio

Time T Cash Flow S(T) < 950

950 ≤ S(T)

Long zeros (face value 1,000)

1,000

1,000

Long 10 calls (strike price 950)

0

Net cash flow

1,000

10[S(T) – 950] 1,000 + 10[S(T) – 950]

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Example 7.1 is a commodity-indexed note, which has its return tied to the performance of a commodity (like platinum, gold, silver, or oil) or a commodity index (like Bridge/CRB Futures Index, a commodity futures index that goes back to 1957). Sometimes, the holder can get unlimited benefits from price rises (as in this example), but at other times, the payoffs are capped. Close cousins of these derivatives are equity-linked notes (ELN). An ELN is a combination of a zero-coupon (or a small coupon) bond whose return is based on the performance of a single stock, a basket of stocks, or a stock index (see Gastineau and Kritzman [1996] for definitions and discussions). These come in numerous variations and have fancy trade names like ASPRINs, EPICs, GRIP, SIR, and SUPER. This extra kicker from the equity component makes these securities attractive to buyers, helps the issuers raise more cash up front, and also hedges some of the inherent risks in their business. Example 7.2 shows a hybrid security created to achieve a tax-efficient disposal of a stock that has appreciated in value.

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EXAMPLE 7.2: Hybrid Securities ■

Suppose you are the chief executive officer of Venturecap Co. (a fictitious name) that has bought 1 million shares in a start-up company Starttofly Inc. at $3 per share. The start-up has done very well and is about to go public. Because Venturecap bought these shares through a private placement, securities laws prevent selling those shares for two years. However, you want to dispose of these shares because finance academics report that high prices achieved after an initial public offering usually disappear after six months.

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What should you do? One alternative is for Venturecap to issue hybrid debt (also called a structured note), which is a combination of a bond and a stock. Your investment banker can design premium equity participating securities (PEPS) that allow you to get the benefits of a sale without actually selling Starttofly’s stock.

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How does this work? Suppose the investment banker designs a three-year bond that pays interest at an annual coupon rate of 4 percent per year on the PEPS selling price of $20. The buyer of each PEPS also gets a share of Starttofly after three years. Many investors like the idea of buying Starttofly for $20 in the future and getting the coupon payments over the intermediate years, and the no-arbitrage principle ensures that this security is equivalent to buying a bond and buying the stock.

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Venturecap sells 1 million PEPS and raises $20 million up front. Moreover, it pays capital gains taxes only when it sells the Starttofly stock, which is three years from now, and it gets to deduct the interest payments of $800,000 from its taxes every year.

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Hybrids have been sold in the US, Asia, Europe, and Australia.

APPLICATIONS AND USES OF SWAPS

7.4

An Introduction to Swaps

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Swaps are one of the most successful uses of financial engineering. A swap is an agreement between two counterparties to exchange (“swap”) a series of (usually) semiannual cash flows over the life (tenor or term) of the contract. Salomon Brothers arranged the first swap between International Business Machines Corp. and the World Bank in 1981. A swap contract specifies the underlying currencies, applicable interest rates, payment timetable, and default contingencies. Being over-the-counter (OTC) instruments, swaps do not enjoy the same level of protection as exchange-traded derivatives. You must carefully document them and, if necessary, add collateral provisions to mitigate credit risk (see Chapter 22 for a detailed discussion of swap documentation). As swaps are customized, they come in many flavors and varieties. The simplest swaps go by the fancy name of plain vanilla, whereas more complex varieties are called exotics. Parties in a plain vanilla interest rate swap exchange fixed for floating interest payments, while those in a plain vanilla currency swap exchange fixed payments in one currency for fixed payments in another currency. The libor rate index (see Chapter 2) has traditionally been used as a reference rate for the cash payments on most swap contracts. Nowadays, a large number of swaps are denominated in the European Union’s euro as well as other major currencies like the Japanese yen and the pound sterling. Swaps have flourished because they provide an efficient way of transforming one cash flow to another with very low transaction costs. Our discussion of how the pursuit of lower transaction costs is shaping the global markets is relevant here (see Extension 1.1).

7.5

Applications and Uses of Swaps

Swaps have many financial engineering uses: ■

Transforming loans. A company can transform a fixed interest rate loan into a floating interest rate loan and vice versa through a plain vanilla interest rate swap.

■

Hedging currency risk. A Japanese manufacturer repatriating profits from its US subsidiary can convert dollars into yen through a cross-currency swap and thereby protect itself from currency risk.

■

Altering asset/liability mix. The 1970s showed that a sharp rise in interest rates can bankrupt a savings and loan bank (S&L) specializing in home mortgage lending. A rate hike increases the value of the liabilities (because the deposits must be paid higher interest) but has little immediate impact on the assets (which primarily consist of fixed rate mortgage loans). An S&L can hedge this risk via an interest swap, in which the S&L pays a fixed rate but receives a floating rate. This swap lowers the mismatch in cash flows from both the S&L’s floating rate liabilities (deposits) and fixed rate assets (mortgage loans).

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Creating payoffs that are hard to attain. As there are no exchange-traded derivatives on jet fuel, an airline can hedge its fuel costs by entering into a commodity swap

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that pays the average aviation fuel price computed over a month in exchange for a fixed payment. ■

Avoiding market restrictions. An international investor who is prevented from investing in a country’s stocks by local governmental rules can go around this restriction by entering an equity swap that pays her an amount tied to the return on the local stock index.

7.6

Types of Swaps

As in other financial markets, the swap market has its own financial intermediaries. These swap facilitators (also called swap banks or just banks) either act as a broker (which is becoming less common these days) or perform the role of a dealer. For simplicity, we omit these dealers from our examples. This hardly alters the story. Just think of these dealers as intermediaries who siphon off a few basis points from the cash flows without affecting the basic nature of the transactions.

Interest Rate Swaps

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A plain vanilla interest rate swap exchanges fixed for floating interest rate payments at future dates. As all payments are denominated in the same currency, a notional principal is used to determine the payments, and cash flows are netted. Example 7.3 shows the setup and cash flows from a plain vanilla interest rate swap (see Chapter 22 for a valuation formula).

EXAMPLE 7.3: A Plain Vanilla Interest Rate Swap (Fixed-for-Floating)

Contract Setup ■

Suppose that Fixed Towers Inc. borrowed $100 million for three years at a floating rate of the one-year libor index rate, but it now wants to switch to a fixed interest rate loan. Floating Cruisers Co. raised the same sum and for the same period at a fixed rate of 6 percent but now wants to switch to a floating rate, the one-year libor rate index.

■

They swap. They work out a deal in which Fixed agrees to pay Floating 6 percent and receive the libor rate index on the $100 million notional. This will go on for three years, which is the swap’s tenor or term. The swap is shown in Figure 7.2.

■

Though swaps often have quarterly payments, let’s assume for simplicity that they exchange cash flows at year’s end. As a result of this agreement, Fixed’s cash flows are (– libor + libor – 6 percent =) – 6 percent times the notional, whereas Floating’s cash flows are (– 6 percent + 6 percent – libor =) – libor times the notional. The swap switched each of their existing loans to the other type.

■

It makes no sense to simultaneously exchange $100 million for $100 million. That’s why $100 million is called the notional principal—it exists as a notion, never changes hands, and is only used for computing payments.

TYPES OF SWAPS

The Cash Flows ■

We compute the net payments assuming that the libor index rate takes the values 5, 6, and 7.50 percent, respectively, over years 1, 2, and 3. (1st year, libor = 5 percent) Fixed owes Floating 6 percent times the notional. Floating owes Fixed 5 percent times the notional. Fixed will pay Floating 1 percent of $100 million or $1,000,000 at the end of the first year. (2nd year, libor = 6 percent) Net payment is 0.

FIGURE 7.2: Example of a Plain Vanilla Interest Rate Swap Transforming a Fixed Rate Loan Into a Floating Rate Loan

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Before Swap

Fixed Towers Raised $100 million Cost libor per year

Floating Cruisers Raised $100 million Cost 6 percent per year

libor

6 percent

After Swap (for three years)

Fixed Towers Net cost 6 percent per year

libor

bbalibor 6 percent

Floating Cruisers Net cost libor per year

6 percent

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(3rd year, libor = 7.50 percent) Fixed owes Floating 6 percent times the notional. Floating owes Fixed 7.50 percent times the notional. Fixed receives from Floating 1.50 percent of $100 million or $1.5 million at the end of the third year. Then the contract ends. ■

In reality, counterparties in swaps and other OTC derivatives contracts sign detailed bilateral agreements. Since its founding in 1985, the International Swap Dealers Association (later renamed the International Swaps and Derivatives Association) has been instrumental in developing convenient documents for these contracts.

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Forex Swaps “Rule, Britannia, rule the waves.”2 After attaining supremacy in the early eighteenth century, the Royal Navy was the world’s most powerful navy for over two hundred years, and the British pound reigned supreme. However, the pound lost some of its luster in the mid-twentieth century when the Bretton Woods system established the dollar as the standard currency. The dollar’s dominance is seen in the foreign exchange (forex) market, where most transactions involve exchange of foreign currencies for dollars, and vice versa. The term exchange rate denotes the price of one currency in terms of another. The currencies are quoted either in direct or American terms as dollars per unit of foreign currency (say, $2 per pound sterling) or in indirect or European terms as foreign currencies per unit of dollar (say, £0.5 per dollar). An exchange rate between two currencies other than US dollars is referred to as a cross-rate. Just as you buy an ounce of gold by paying dollars, think of foreign currencies as tradable goods with dollar prices. If the price goes up and you have to spend more dollars to buy one unit of a foreign currency (say, £1 = $3), then we say that the dollar has depreciated. Fewer dollars for a unit of foreign currency (say, £1 = $1) means that the dollar has appreciated. Forex swaps are widely used in forex markets for managing currency risks. A forex swap (or FX swap) is a single transaction with two legs: (1) the contract starts with a spot exchange of one currency for another and (2) it ends with a reverse exchange of these currencies at a subsequent date. Example 7.4 shows the basic idea behind a forex swap.

2

“Rule, Britannia!” is a patriotic British national song, which originated from the poem with the same name written by Scottish poet James Thomson in the mid-eighteenth century.

TYPES OF SWAPS

EXAMPLE 7.4: A Forex (FX) Swap ■

Suppose that Americana Bank has $200 million in excess funds for which Britannia Bank (both fictitious names) has an immediate need. They enter into a forex swap with a tenor of one month.

■

The spot exchange rate SA is $2 per pound sterling in American terms, and its inverse SE = 1/2 = £0.50 per dollar in European terms. Americana gives $200 million to Britannia and receives (Dollar amount × Spot exchange rate in European terms) = $200 × £0.50 per dollar = £100 million today.

■

The annual simple interest rates are i = 6 percent in the US and iE = 4 percent in the United Kingdom. After one month (using a monthly interest rate computation for simplicity), Britannia repays Americana, Dollar amount × Dollar return = $200 × [1 + (0.06/12)] = $201 million and Americana repays Britannia,

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Foreign currency amount × Value of one pound invested for one month = £100 × [1 + (0.04/12)] = £100.3333 million

Instead of beginning immediately, a forex swap may start at a future date. A majority of forex swaps have lives that are measured in weeks rather than months. These versatile instruments have many uses such as (1) managing cash flows in connection with imports and exports, (2) managing borrowings and lendings in foreign currencies, (3) handling foreign exchange balances, and (4) temporarily exchanging excess amounts of one currency for another.

Currency Swaps A plain vanilla currency swap (or a cross-currency swap) is an arrangement between two counterparties involving (1) an exchange of equivalent amounts in two different currencies on the start date, (2) an exchange of interest payments on these two currency loans on intermediate dates, and (3) repayment of the principal amounts on the ending date. Because it’s an OTC contract, the counterparties can modify the terms and conditions at mutual convenience. Some currency swaps skip the exchange of principal and just use them as notional amounts for computing interest. Interest payments are made in two different currencies and are rarely netted.

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Corporations use currency swaps to exchange one currency into another. They do this to repatriate profits and to set up overseas factories. Example 7.5 shows how such a swap works.

EXAMPLE 7.5: A Plain Vanilla Currency Swap (Fixed-for-Fixed) ■

Suppose that Americana Auto Company wants to build an auto plant in the UK and Britannia Bus Corporation (both fictitious names) wants to do the same in the US. To open a factory, you need local currency. Often a manufacturer can more easily raise cash at home because it has relationships with local banks. Americana and Britannia both plan to do this, and seeing a large spread between foreign exchange buying and selling rates, they decide to do the currency conversion via a currency swap. This is a generic fixed-for-fixed currency swap that involves regular exchange of fixed payments over the swap’s life.

The Swap Contract ■

The automakers enter into a swap with a three-year term on a principal of $200 million. The spot exchange rate SA is $2 per pound. Americana raises 100 × 2 = $200 million and gives it to Britannia, which, in turn, raises £100 million and gives it to Americana. Assuming annual payments, Americana pays Britannia 4 percent on £100 million and Britannia pays Americana 6 percent on $200 million at the end of each year for three years. The companies are basically exchanging their borrowings.

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■

The swap ends after three yearly payments, and the principals are handed back. Figure 7.3 shows these cash payments.

Extension 7.1 provides a valuation formula for fixed-for-fixed currency swaps. Currency swaps have several interesting features. First, a swap’s value depends on two sets of interest rates: domestic as well as foreign interest rates of different maturities.3 Second, although our example assumes annual payments, most swaps involve semiannual payments. You can easily add this feature. Notice that Result 7.1 of Extension 7.1 is quite general and does not depend on whether we have annual or semiannual cash flows. Third, the swap value is arbitrage-free. If an errant dealer quotes a different price for the same swap, then you can buy “the cheaper and sell the more expensive swap” and create arbitrage profits. Fourth, anything goes in the swaps markets—these OTC contracts can be designed to satisfy the whims and fancies of the counterparties.

3

Using the covered interest rate parity formula (see Result 12.3), you can eliminate the foreign interest rate and rewrite the formula in terms of the spot and the currency forward rate.

TYPES OF SWAPS

FIGURE 7.3: Plain Vanilla Currency Swap Today: Swap Begins with Exchange of Principal

$200 million Americana Raises $200 million in US at 6 percent

£100 million

Britannia Raises £100 million in UK at 4 percent

$200 million US investors

£100 million UK investors

Intermediate Dates: Interest Payment at the End of First and Second Years

$12 million

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Americana

£4 million

Britannia

$12 million US investors

£4 million UK investors

Maturity Date: Interest Payment and Principal Repayment

$212 million Americana

$212 million US lenders

£104 million

Britannia

£104 million UK lenders

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EXTENSION 7.1: Valuing Fixed-for-Fixed Currency Swaps We expand upon Example 7.5 to develop a valuation formula for currency swaps.

EXT. 7.1 EX. 1: A Valuation Formula for a Currency Swap

Contract Setup ■

■

The situation and the terms guiding this swap are as follows: - Consider the US dollar as the domestic currency and pounds sterling as the foreign currency. - The swap’s tenor T is three years, and it involves annual interest payments. - The spot exchange rate SA is $2 per pound in American terms, which is SE = 1/SA = £0.50 per dollar in European terms. - Americana raises the principal amount L = $200 million at the coupon rate i of 0.06 or 6 percent per year. - Britannia raises an equivalent sum at the coupon rate iE of 0.04 or 4 percent per year. The principal amount LE is calculated by multiplying the dollar principal by the spot exchange rate: L × SE = LE = 200 × 0.50 = £100 million. Americana pays US investors:

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C = Principal (in domestic currency) × Interest rate (on dollar principal) =L×i = 200 × 0.06 = $12 million at the end of each year for three years. - Britannia pays UK investors an annual amount: CE = Principal (in “European” currency) × Interest rate (on foreign currency principal) = LE × iE = 100 × 0.04 = $4 million

The Cash Flows ■

The swap generates the following cash flows (see Figure 7.3): Time 0: Today The swap begins with an exchange of the principals. Americana pays L = $200 million (shown with a solid line arrow) and gets LE = £100 million from Britannia (shown with a broken line arrow).

TYPES OF SWAPS

143

Time 1 (after one year) and Time 2 (at the end of the second year) Interests are paid at year’s end. Americana pays CE of £4 million to Britannia (see solid line arrow), which passes it on to the original British investors (see broken line arrow). Britannia pays C of $12 million to Americana (broken line arrow), which passes it on to the original American investors (solid line arrow). The companies have exchanged their borrowings. Americana has transformed a dollar liability into a sterling liability, while Britannia has done the opposite and accepted a dollar loan. Time 3 (after three years) The swap ends when the companies return their principals and final interest payments. Americana pays £4 million in£4 million in interest and repays the principal of £100 million to Britannia and receives $212 million.

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A Pricing Model ■

Today’s cash flows cancel and do not affect the swap valuation. As Americana will be receiving dollars and paying sterling, we can value this swap to Americana by adding up the present value of all the dollar payments that the company receives and subtracting from this sum the present value (in dollar terms) of all the sterling payments that the company makes.

■

To compute present values, first find the zero-coupon bond (or “zero”) prices in the two countries. They come from the government securities in the US and the UK. The UK government–issued bonds are called gilt securities. Like their American cousins, gilts have a whole range of maturities, semiannual coupon payments, STRIP features, callable issues, and the presence of inflation-protected securities. The UK has even issued consols, which are perpetual bonds that never mature.

■

Denote US zero prices by B(t) and UK zero prices by B(t)E , where t = 1, 2, and 3 years are the times to maturity (see Ext 7.1 Tab. 1).

■

Discount the cash flows by multiplying them by the respective zero-coupon bond prices (Ext. 7.1 Tab. 1 shows how this works). Multiplying the dollar receipts by the US zero prices, we get the following: Present value of Americana’s dollar receipts = PV of dollar interest payments + PV of dollar principal = [(Price of a US zero − coupon bond maturing in one year × Interest payment) + (Price of a zero − coupon bond maturing at time 2 × Cash flow) + (Price of a zero − coupon bond maturing in year 3 × Interest)] + (Price of a zero − coupon bond maturing in three years × Principal) = [B (1) C + B (2) C + B (3) C] + [B (3) L] = (0.95 + 0.89 + 0.82) × 2 + (0.82 × 200) = $195.92 million = PV

(7.1)

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EXT. 7.1 TABLE 7.1: Zero-Coupon Bond Prices in the United States (Domestic Country) and the United Kingdom (Foreign Country) Time to Maturity (in years)

US (Domestic) Zero-Coupon Bond Prices (in dollars)

UK (Foreign) Zero-Coupon Bond Prices (in pounds sterling)

1

B(1) = $0.95

B(1)E = £0.96

2

B(2) = 0.89

B(2)E = 0.91

3

B(3) = 0.82

B(3)E = 0.85

Multiplying the sterling payments by UK zero prices, we get the present value of Britannia’s sterling receipts, which equals Americana’s payments. Consequently, Present value of Americana’s sterling payments = PV of interest payments in pounds sterling + PV of sterling principal = [B (1)E CE + B (2)E CE + B (3)E CE ] + [B (3)E LE ] = (0.96 + 0.91 + 0.85) × 4.00 + 0.85 × 100 = £95.88 million = PVE

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■

Convert this into US dollars by multiplying it by the spot exchange rate: Present value of Americana’s sterling payments in terms of today’s dollars = PVE × SA = 95.88 × 2 = $191.76 million

■

(7.2)

(7.3)

The dollar value of the foreign-currency swap to Americana Auto Company is Value of the currency swap to the domestic investor = (Present value of Americana’s dollar receipts) − [Present value of Americana’s sterling payments (in dollar terms)] = PV − PVE × SA = 195.92 − 191.76 = $4.16 million This means Americana must pay Britannia $4.16 million to enter into the swap.

(7.4)

TYPES OF SWAPS

■

Owing to credit risk, it is prudent to avoid any upfront payment and make this a par swap, which has a zero value. You can do this by tweaking the interest rates until this happens. For example, if you keep i fixed, you will find that iE = 0.047647 makes this a par swap. We restate this as a result (see Ext 7.1 Result 1).

EXT. 7.1 RESULT 1 Valuing a Plain Vanilla Currency Swap A currency swap begins at time 0, when 1. a domestic counterparty (“American”) pays the principal L dollars to a foreign counterparty (“European” or “Foreigner”) and receives an equivalent sum LE in foreign currency 2. American agrees to pay a fixed interest on LE to Foreigner in exchange for a fixed dollar interest on L at the end of each period (times t = 1, 2, . . . , T) 3. the counterparties repay the principals on the maturity date T Then, The value of the swap to American, VSWAP = Present value of dollar receipts − Present value of foreign currency payments (converted dollars into US at the spot rate) = PV − PVE × SA

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145

(7.5)

where PV is the present value of future dollar cash flows received by American, PVE is the present value of future foreign currency payments, and SA is today’s spot exchange rate in American (dollar) terms.The present value of the cash flows is computed as T

PV =

B (t) C + B (T) L

(7.6)

B (t) i + B (T) L ]

(7.7)

∑ t=1

T

=

[∑ t=1

where B(t) is today’s price of a US zero-coupon bond that pays one dollar at time t and C is the coupon or the interest on the dollar principal that is paid at times t = 1, 2, . . . , T (computed as dollar principal times the simple interest rate or L × i). PVE is similarly computed by attaching a subscript E and by replacing domestic by foreign currency cash flows and zero-coupon bond prices in the preceding formula.

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Commodity and Equity Swaps Example 7.6 illustrates a simple commodity swap.

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EXAMPLE 7.6: Hedging Airline Fuel Costs with a Commodity Swap ■

Aviation fuel is a major cost for airlines. Some airlines hedge this price risk by entering into a commodity swap in which they pay a fixed price and receive the average aviation fuel price computed over the previous month.

■

Suppose HyFly Airlines (fictitious name) needs 10 million gallons of aviation fuel per month, which it buys on a regular basis from the spot market. Seeing a huge rise in oil (and aviation fuel prices) in recent years, HyFly is scared that a continued run-up will ruin the airline, and it decides to hedge its exposure by entering into a commodity swap.

■

The swap is structured as follows (see Figure 7.4 for a diagram of this swap): - The notional is 10 million gallons of aviation fuel. - The swap has a two-year term, and the payments are made at the end of each month. - HyFly pays the dealer a fixed price of $1.50 per gallon or 10,000,000 × 1.50 = $15 million each month. - The dealer pays a floating price that is the average spot price of fuel during the previous month. - The payments are net.

■

Suppose last month’s average price is $1.60 per gallon. - Then the dealer pays 10,000,000 × 1.60 = $16 million at the end of the this month. - As the payments are netted, the airline receives $1 million. - This extra payment helps alleviate the airline’s higher aviation fuel price paid in the spot market.

■

Suppose oil prices fall unexpectedly and last month’s average fuel price was $1.30. The savings that HyFly generates from the lower price have to be surrendered to the dealer, who must be paid $2 million$2 million by the terms of the contract.

This commodity contract is hard to price because it involves an average price (see Jarrow and Turnbull [2000] for a valuation formula for this swap). Equity swaps are similar to the swaps previously discussed. The simplest equity swaps involve counterparties exchanging fixed rate interest payments for a floating payment that is tied to the return on a stock or a stock index. The payments are computed on a notional and exchanged at regular intervals over the swap’s term. The index may be a broad-based index (such as the Standard and Poor’s 500) or a narrowly defined index (such as for a specific industry group like biotechnology). Equity swaps allow money managers to temporarily change the nature of their portfolios and bet on the market. Suppose that we find two portfolio managers who hold completely opposite views about the direction of the stock market. They can enter into an equity swap in which the bullish manager receives a floating rate tied to the market index and pays the bearish manager a fixed interest rate.

TYPES OF SWAPS

FIGURE 7.4: Hyfly Airline Hedging Fuel Price Risk For 2 Years

$1.50 per gallon Hyfly Airline

Swap Dealer Average spot price

Aviation fuel

Spot price of aviation fuel

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Spot Market

Credit Default Swaps Credit default swaps (CDS) have been hailed during good times as a “wonder of modern finance” and vilified during the financial crisis of 2007 as a “‘Ponzi scheme’ that no self-respecting firm should touch” (see “Credit Derivatives: The Great Entangling,” Economist, November 6, 2008). Designed during the 1990s (by J. P. Morgan & Co.), CDS combine the concepts of insurance and swaps to create a contract that plays a useful economic role. It enables the holder of a risky bond to sell the credit risk embedded in the bond to a counterparty willing to bear it. Let’s consider how a CDS works. A CDS is a term insurance policy on an existing corporate or government bond. A typical CDS matures in one to five years and has a notional amount equal to the face value of the bond being insured, with most CDS being in the $10 to $20 million range. The seller of a CDS (the insurer) receives a periodic premium payment (usually quarterly) for selling the insurance. If a credit event occurs—the bond defaults—then the buyer of the CDS (the insured) receives the face value of

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the debt from the seller but hands over the bond in return.4 The CDS contract ends if a credit event occurs. The premium payment is quoted as a CDS spread (say, 100 basis points per annum). The actual dollar payment is the prorated spread (adjusted for the length of the payment period) times the notional. The CDS spread is quoted in the market so that no cash is exchanged when the CDS is written. This implies that the CDS has zero value at initiation. Example 7.7 illustrates the working of a CDS.

EXAMPLE 7.7: Hedging Default Risk ■

A pension fund, the Worried Lenders Fund (a fictitious name) has purchased a bond issued by Candyfault Enterprises (CE) (another fictitious name). The bond has five years until it matures, pays a 5.5 percent annual coupon, and has a notional of $5 million. The pension fund manager is concerned that Candyfault might default on its interest payments over the next year owing to the current recession. She wants to hedge its default risk but does not want to sell the bond.

■

To hedge the default risk on CE over the next year, it buys a one-year CDS on CE with a notional of $5 million. The CDS quoted spread is 400 basis points per year, paid quarterly. The Worried Lenders Fund buys this CDS.

■

The quarterly premium payment made by the Worried Lenders Fund is 0.04 × (1/4) × $5 million = $50, 000

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■

The CDS seller gets this payment quarterly until the CDS expires or until CE defaults on its bond, whichever comes first. In the event of default, the CDS seller pays the Worried Lenders Fund $5 million, and it takes possession of the CE bond.

Pricing of this simple contract, however, is a difficult task. CDS are also written on asset backed securities, called ABS CDS, but their cash flows are somewhat different (see Jarrow 2011).

7.7

Summary

1. We emphasize a build and break approach, in which derivatives can be broken down into component parts or combined with other derivatives to create new ones. 2. Once the building blocks are understood, one can venture into financial engineering, a new discipline that applies engineering methods to financial economics. 3. A swap is an agreement between two counterparties to exchange a series of cash payments over its life. Swaps are successful vehicles for transforming one kind of 4

This is the case of physical delivery. If cash delivery occurs instead, the swap buyer receives the difference between the face value of the debt and the market price of the defaulted debt issue (usually determined through an auction).

QUESTIONS AND PROBLEMS

cash flow to another. Swaps are OTC contracts that can be customized to the counterparties. The simplest kinds of swaps have the following structure: a. A plain vanilla interest rate swap exchanges fixed for floating rate payments. The principal, called the notional principal, never changes hands but is used only for computing cash flows. Payments are netted because the cash flows are denominated in the same currency. b. A forex swap involves a spot exchange of foreign currencies that is followed by a reverse exchange of equivalent amounts (principal plus relevant interest) at a future date. c. A plain vanilla currency swap (or a cross-currency swap) is an arrangement between two counterparties involving (1) an exchange of equivalent amounts of two different currencies on the starting date, (2) an exchange of interest payments on these two currency loans on intermediate dates, and (3) repayment of the principal on the ending date. d. A commodity swap involves an exchange of an average price of some notional amount of a commodity in exchange for a fixed payment. e. An equity swap exchanges fixed rate interest payments for a floating payment that is tied to the return on a stock or a stock index. f. A credit default swap is a term insurance policy on a bond.

7.8

Cases

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Times Mirror Company PEPS Proposal Review (Harvard Business School Case

296089-PDF-ENG). The case examines the design of a premium equity participating security for tax-efficient disposal of an appreciated common stock. Privatization of Rhone-Poulenc—1993 (Harvard Business School Case 295049-

PDF-ENG). The case illustrates how a recently privatized French company can use financial engineering to design an investment plan for employees that would enhance their ownership of company shares. Advising on Currency Risk at ICICI Bank (Harvard Business School Case 205074-

PDF-ENG). The case studies how a large Indian bank can hedge risks from multiple interest rates and currencies by using a variety of derivatives like swaps, options, and futures contracts on interest rates and currencies.

7.9

Questions and Problems

7.1. Briefly describe a plain vanilla interest rate swap. Is the notional principal paid

out at the swap’s beginning and maturity? 7.2. Briefly describe a currency swap. Is the notional principal paid out at the swap’s

beginning and maturity? 7.3. How does a plain vanilla interest rate swap differ from a currency swap?

The next two questions use the following information: American Auto enters into a plain vanilla currency swap deal with European Auto. American raises $30 million at an 8 percent fixed interest rate, while European raises an equivalent amount of €33 million at 9 percent. They swap

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these two loans. The swap lasts for three years, and the payments take place at the end of each year. 7.4. What is the principal? Does it change hands at the beginning and end of the

swap? 7.5. Calculate the gross payments involved and indicate who pays what in this swap

deal. The next two questions use the following information: Esandel Bank enters into a plain vanilla interest rate swap with a swap facilitator Londoner Inc. Esandel pays a fixed amount of 8 percent per year to Londoner, which in turn pays a floating amount libor + 1.50 percent. The principal is $15 million. The swap lasts for five years, and the payments take place at the end of each year. 7.6. What is the notional principal? Does it change hands at the beginning and end

of the swap? 7.7. Who is in the “receive fixed” situation? Who is in the “pay-fixed” situation? 7.8. Calculate the net payments involved and indicate who pays what in this swap

deal if the libor takes on the values 7.00 percent, 6.50 percent, 7.00 percent, 7.50 percent, and 6.00 percent at the end of the first, second, third, fourth, and fifth year, respectively. Consider the swap in Extension 7.1:

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- The automakers enter into a swap with a three-year term on a principal of $200 million. - The spot exchange rate SA is $2 per pound. Americana raises 100 × 2 = $200 million and gives it to Britannia, which in turn raises £100 million and gives it to Americana. - Americana pays Britannia 4 percent per year on £100 million and Britannia pays Americana 6 percent per year on $200 million at the end of each year for three years. Now assume that the companies make payments every six months: the swap ends after six semiannual payments, and the principals are handed back after three years. 7.9. Using zero-coupon bond prices (maturing every six months) given in

Table 7.3, compute the value of this swap. 7.10. How can one use a currency swap to hedge currency risk? 7.11. How can one use an interest rate swap to change a fixed rate loan into a floating

rate loan? 7.12. How can a savings and loan bank use an interest rate swap to match its long-

term fixed rate investments with the risks of its short-term floating rate demand deposit obligations?

QUESTIONS AND PROBLEMS

TABLE 7.3: Zero-Coupon Bond Prices in the United States (Domestic Country) and the United Kingdom (Foreign Country) Time to Maturity (in years)

US (Domestic) Zero-Coupon Bond Prices (in dollars)

UK (Foreign) Zero-Coupon Bond Prices (in pounds sterling)

0.5

B(0.5) = $0.99

B(0.5)E = £ 0.98

1

B(1) = 0.97

B(1)E = 0.96

B(1.5) = 0.95

B(1.5)E = 0.93

B(2) = 0.93

B(2)E = 0.91

B(2.5) = 0.91

B(1.5)E = 0.88

B(3) = 0.88

B(3)E = 0.85

1.5 2 2.5 3

7.13. What is an equity swap? What is a commodity swap? 7.14. Suppose an investor is precluded from investing in a country’s stocks by

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government regulations but can invest in the country’s government bonds. Design a swap to overcome these regulations. 7.15. What is a forex swap? Explain how it works. 7.16. In Chapter 2, we described how the US Treasury STRIPS worked. Explain

how Wall Street firms had financially engineered a product similar to STRIPS. 7.17. What is a callable bond? When would it be used? 7.18. What is a putable bond? When would it be used? 7.19. What is a convertible bond? How would you break it down into simpler parts

and price it? 7.20. Check the Internet or other sources and answer the following questions: a. Briefly describe the Troubled Asset Relief Program (TARP) program of the

US government. b. What was the derivative used in the program? Why would the US

government trade such a “toxic thing” as a derivatives contract? c. How would you approach the pricing of the security used in the TARP? d. Discuss the success of the program to date.

151

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II

Forwards and Futures

CHAPTER 8

Forwards and Futures Markets

CHAPTER 9

Futures Trading

CHAPTER 10

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Futures Regulations

CHAPTER 12

The Extended Cost-of-Carry Model

CHAPTER 13

Futures Hedging

CHAPTER 11

The Cost-of-Carry Model

8 Forwards and Futures Markets Copyright © 2018. World Scientific Publishing Company Pte. Limited. All rights reserved.

8.1 Introduction 8.2 Applications and Uses of Forwards and Futures 8.3 A Brief History of Forwards and Futures Early Trading of Forward and Futures-Type Contracts

8.4 Futures Contract Features and Price Quotes Commodity and Financial Futures Contracts The Gold Futures Contract Gold Futures Price Quotes The Exchange and Clearinghouse

US Futures Exchanges and the Evolution of the Modern Futures Contract, 1848–1926

8.5 Commodity Price Indexes

Recent Developments, 1970 Onward

8.6 Summary

EXTENSION 8.1 Doomsters and Boomsters

8.7 Cases 8.8 Questions and Problems

APPLICATIONS AND USES OF FORWARDS AND FUTURES

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8.1

Introduction

In 1688, the world’s first futures exchange, the Dojima Rice Exchange, opened its doors. It was located in Osaka, which was Japan’s commercial center at the time known as the “country’s kitchen.” In those days, if you controlled rice, then you controlled the Japanese economy. The Dojima Exchange devised a fairly advanced system of trading that has many commonalities with the way futures trade today. A day’s trading time was determined by a firebox system. The market would open in the morning with the lighting of a wick in a hanging wooden box, and it would close when the wick was completely burned down. If the wick was extinguished early, then all trades of that day were canceled. You can imagine how losing traders tried to puff out the fire early, while winning traders and exchange officials attempted to foil such plans. Raw emotions would sometimes get expressed through brawls and fistfights (see Alletzhauser 1990; West 2000). As this story relates, emotions (and stakes) can run high at futures exchanges, the workings of which are described in this chapter. However, as floor trading is being phased out, action these days takes place behind electronic terminals in airconditioned rooms. We take up forward and futures contracts from where we left off in Chapter 4. We discuss the usefulness of forward and futures contracts and present a brief history. Although futures-type contracts have traded in many times and places, the evolution of the modern futures contract really began in mid-nineteenth-century Chicago and was completed in about seventy-five years. After decades of relative calm, the 1970s saw a period of astonishing changes in futures markets. These developments can be classified into two categories: (1) the introduction of new futures contracts, and (2) the opening of new exchange and the automation of trading. Finally, we examine a gold futures contract and study how futures prices get reported in the financial press and quoted by vendors.

8.2

Applications and Uses of Forwards and Futures

As forward and futures markets are growing by leaps and bounds (and huge securities markets don’t exist without good reason), they must be serving a useful role in the economy. And this growth is not just in futures on commodities like corn and gold. The 1970s saw the introduction of financial futures (“financials”), which are futures contracts based on financial assets (such as foreign currencies) and on economic or financial variables (such as interest rates). These contracts have become very popular. What is the utility of forward and futures contracts including those written on financial assets and financial variables? Futures and forwards take investors far beyond the realm of ordinary stocks and bonds and, in the language of Star Trek, “where no man has gone before.” Consider the following observations: 1. By purchasing forwards and futures in advance, a trader can smooth unexpected future price fluctuations that can come from demand–supply mismatches.

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2. Forwards are useful for acquiring a commodity at a fixed price at a later date. Futures are seldom used for trading assets because of the costs involved with physical delivery. However, their features make them excellent tools for managing price risk. 3. Forwards and futures help make the market more “complete.” In a complete market, sufficient securities trade such that investors can construct portfolios to obtain all possible probability distributions over future payoffs. 4. They help traders to speculate. Speculators have a beneficial presence in most futures markets. Speculators increase liquidity and enhance market efficiency, but destabilizing speculation can also take place, and we look at several manipulation cases in Chapter 10. 5. They allow investors to leverage their capital and hold large trading positions without tying up cash. However, like speculation, leverage cuts both ways and can sometimes do more harm than good. For example, futures bets destroyed Barings PLC, a venerable British bank that had existed for over two hundred years. 6. The process of trading forwards and futures generates useful information about future price expectations called price discovery, an important function discussed in greater detail in chapter 10.

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8.3

A Brief History of Forwards and Futures

Forward trading began early in the dawn of civilization (see Table 8.1) and thrives to this day. The history of forwards and exchange-traded futures may be divided into three phases: 1. Early trading of forward and futures-type contracts. Forward contracts evolved separately in many parts of the world. Sometimes they traded in centrally located markets with standardized trading rules and contract terms that prompted some scholars to label them as futures contracts. 2. Evolution of the modern futures contract. Today’s futures contracts started taking shape in the United States during the nineteenth century and arrived at its present form over the next seventy-five years. 3. Developments since the 1970s. An astonishing variety of futures were conceived and marketed during the last three decades of the twentieth century.

Early Trading of Forward and Futures-Type Contracts Forward contracts have been used to reduce risk for eons. Forwards traded in India around 2000 BC, in ancient Greece and Rome, in medieval Europe, and in Japan during the seventeenth century (see Table 8.1). Records indicate that Amsterdam grain dealers used both futures and options in the 1550s and the decades that followed to hedge trade goods shipped far distances.

A BRIEF HISTORY OF FORWARDS AND FUTURES

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TABLE 8.1: Early History of Trading Forward and Futures-Type Contracts Year

Development

2000 BC and onward

Forward contracts traded in Greece, India, and the Roman Empire

Eleventh and twelfth centuries AD and onward

Medieval fairs

1531

The first commodities exchange in Antwerp, Belgium

1550s onward

Futures and options contracts traded in Amsterdam, Netherlands

1571

Royal Exchange for trading commodities in London, England

1688

Dojima Rice Exchange in Osaka, Japan

One of the earliest commodities exchanges was the Royal Exchange of London, which was inaugurated by Queen Elizabeth I in 1571. Today the original location houses an upscale shopping mall. The renowned London Metal Exchange and the London Stock Exchange both trace their ancestry back to the Royal Exchange. The world’s oldest stock exchange, as noted in Chapter 2, opened in Amsterdam in 1602. Soon futures and options contracts started trading. This was a time when the tulip was gaining popularity in Holland. A few decades later, the Dutch got involved in an extraordinary speculative fervor over rare tulip bulbs, and futures and options were devised to help the speculating public. As with other price bubbles, the famous tulip bulb price bubble ended when prices came crashing down and derivatives trading dried up. As noted at the beginning of this chapter, Osaka’s Dojima Rice Exchange opened its doors in 1688. It is considered the world’s first organized futures market because the contracts were standardized, had fixed life, required margin payments, and involved clearinghouses. The exchange operated for centuries and was closed down in 1939 because of wartime controls.

US Futures Exchanges and the Evolution of the Modern Futures Contract, 1848–1926 Despite interesting developments elsewhere, modern futures trading really began in Chicago in the mid-nineteenth century with the establishment of the Chicago Board of Trade (CBOT or CBT). Why Chicago? Located on the windswept shores of Lake Michigan, the “Windy City” is close to the fertile farmlands of the great Midwest. Nineteenth-century Chicago became America’s major transportation and

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TABLE 8.2: Evolution of Today’s Futures Contract (1848–1926) Year

Development

1848

The CBOT was established to standardize grains trading

1849–50

To arrive contracts started trading at the CBOT

1865

General rules were developed at the CBOT that standardized futures contracts; traders were required to post margins

1874

The Chicago Produce Exchange was formed to trade farm products; it became the Chicago Butter and Egg Board in 1898, and in 1919, it became the CME, with its own clearinghouse

1877

Speculators were allowed to trade at the CBOT

1882

NYMEX was established

1925

A clearinghouse was established at the CBOT

distribution center for agricultural produce. Farmers shipped their grains to Chicago, much of which then moved eastward to where most Americans lived. But agriculture is a seasonal business. Prices crashed at harvest times as farmers flooded Chicago with grains and rose again as the grain was used and became scarce. A need was felt for a central marketplace to smooth such demand–supply imbalances. The futures market was devised to fill this void. To standardize the quality and quantity of grains traded, eighty-two merchants founded the CBOT in 1848. Spot market trading began immediately, and more developments followed. Table 8.2 shows how US futures trading evolved to reach its present-day perfection. In the mid-1860s, the CBOT streamlined trading and further standardized contracts in terms of size, quality of commodities, delivery dates, and places. The contracts came to be called futures. Most important, the CBOT also required performance bonds called margins from both buyers and sellers. This eliminated counterparty risk. In 1877, the CBOT allowed speculators to trade. Prior to this time, only producers or purchasers could trade. In contrast, speculators trade for profits and not for hedging purposes. Meanwhile there was a need to organize the trading of butter, eggs, poultry, and other farm products. Merchants trading these commodities founded the Chicago Produce Exchange in 1874; this became the Chicago Butter and Egg Board in 1898 and emerged as the Chicago Mercantile Exchange (CME or the Merc) in 1919. It became widely known for trading traditional futures contracts on dairy, meat, and poultry products. A group of merchants founded the Butter and Cheese Exchange of New York, renamed the New York Mercantile Exchange (NYMEX) in 1882. Another innovation came when clearinghouses were established by the CME after its reorganization in 1919, and by the CBOT in 1925. As noted earlier, in exchangetraded derivatives markets, a clearinghouse plays the crucial role of clearing a trade by matching the buyer and seller, recognizing and recording trades, and (nearly)

A BRIEF HISTORY OF FORWARDS AND FUTURES

159

eliminating counterparty default risk. These developments almost created the futures contract as we know it today.

Recent Developments, 1970 Onward Futures markets have undergone significant changes since 1970 (see Table 8.3). We classify these changes into two categories: (1) the introduction of new futures contracts, and (2) the opening of new exchanges and the automation of trading.

NEW FUTURES CONTRACTS In 1972, the CME created the International Monetary Market (IMM) division for trading forex futures. The seven foreign currency futures that started trading at the IMM became the first successful financial futures traded. But exchanges don’t always succeed. Credit for the first currency futures exchange goes to the International Commercial Exchange, an exchange founded by the members of the New York Produce Exchange in 1970 but that closed its doors in 1973. Its failure has been attributed to several causes: (1) bad timing (as it was founded when the Bretton Woods agreement was still in force), (2) limitations of a small exchange, (3) small contract size, (4) a failure to adequately promote itself, (5) high margin requirements, and (6) high brokerage fees (see “New Game in Town,” Wall Street Journal, May 16, 1972).

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TABLE 8.3: Milestones in the History of Futures Trading since 1970 Year

Development

1972

The first successful financial futures, foreign currency futures, started trading at the newly created IMM at the CME

1973

The CBOE was founded by members of the CBOT

1974

US Congress passed the Commodity Futures Trading Commission Act that created the CFTC

1975

The first interest rate futures, GNMA futures contracts, started trading at the CBOT

1977

Treasury bond futures began trading at the CBOT

1981

The first cash-settled contract, the eurodollar futures, started trading at the CME

1982

The NFA, a self-regulatory organization made up of firms and people who work in the futures industry, was established

1982

Options on US Treasury bond futures started trading at the CBOT

1982

The first index futures contract, the Value Line index futures, started trading at the Kansas City Board of Trade; within months, the S&P 500 stock index futures started trading at the CME

1984

The first international futures link, the CME/SIMEX mutual offset trading link, was established

1992

A postmarket global electronic transaction system, Globex, was launched by the CME and Reuters

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CHAPTER 8: FORWARDS AND FUTURES MARKETS

Novel products and new markets generate a need for greater regulatory oversight. In 1974, the US Congress passed the Commodity Futures Trading Commission (CFTC) Act, which created the CFTC as the “federal regulatory agency for futures trading.” In 1982, the National Futures Association (NFA), an industry body, was established. NFA does a slew of self-regulatory activities, which (remember from Chapter 1) enhances the reputation of the futures markets and reduces the need for greater federal oversight. In the 1970s, oil shocks and other supply-side disturbances hiked up the inflation rate, causing interest rates to be more volatile. In 1975, the CBOT launched the Government National Mortgage Association (GNMA) futures, which protected mortgage holders from interest rate risk. This was the first interest rate derivative. The Ginnie Mae futures contract was the spark that ignited the creation of the vast global market for interest rate derivatives, one of which was the eurodollar futures introduced by the CME in 1981. Eurodollar futures were the world’s first cash-settled futures contracts. Normally, when a futures contract is carried to delivery, the short collects the futures price and delivers the underlying commodity. A cash-settled futures contract has no such physical delivery provision. If taken to maturity, a cash payment equal to the difference between the settlement price at the contract’s end and the previous day’s settlement price closes out the contract. Without cumbersome delivery, it became possible to design an extraordinary range of derivatives based on notional variables like indexes, interest rates, and other intangibles. The year 1982 saw the introduction of options both on futures and on stock index futures. In Chapter 5, we introduced options on spot, which are options contracts based on an underlying asset or a notional variable. Equity options on blue chip stocks like Ford and International Business Machines are its simplest examples. By contrast, options on futures or futures options have a futures contract as the underlying. This market started in 1982 with US Treasury bond futures. Most futures options now trade side by side with their underlying futures contracts on their respective exchanges. Index futures came as a natural next development. These are cash-settled futures contracts that have an index as the underlying variable. In 1982, the Kansas City Board of Trade it got regulatory approval to trade futures on the Value Line Index. This is an average of over 1,700 stocks tracked by Value Line Incorporated, a well-known provider of investment advice. Months later, the CME introduced the Standard and Poor’s 500 index futures (S&P 500), which soon overtook the former. Index futures are very popular for implementing different trading strategies, including hedging, speculation, and index arbitrage. Subsequent years saw the introduction of an assortment of new contracts. Futures contracts on interest rates in major currencies like dollars (eurodollars), the euro, the Japanese yen, and the pound sterling are offered by major exchanges including the CME Group. Moreover, they trade on short-term interest rate benchmarks, which are often determined by a method similar to the libor index rate (see Chapter 2) and use the suffix IBOR or BOR (for “interbank offered rate”). Examples include interest rate futures on BUBOR in Budapest, HIBOR in Hong Kong, KORIBOR

FUTURES CONTRACT FEATURES AND PRICE QUOTES

in Korea, MOSIBOR in Moscow, and STIBOR in Stockholm. The emergence of the European Community has also created the need for euro-based futures. Just open the Wall Street Journal or go to the website of one of the futures exchanges to get an idea of this diversity.

NEW EXCHANGES AND THE AUTOMATION OF TRADING In terms of trading volume, some of the world’s largest futures and options exchanges have opened in recent years. There are futures exchanges even in countries with emerging financial markets. In all, nearly a hundred futures exchanges operate today. Many of them are modeled after US futures exchanges, and they even relied on American technical expertise to be set up. The European Union’s motto “United in Diversity” is an apt description of today’s financial world: numerous exchanges with diverse product lines in distant lands have proliferated with customers linked through a single electronic terminal. Recent years have seen a trend toward extended trading hours. As discussed in Chapter 3, IT advances have revolutionized securities trading by making innovations like electronic exchanges, online brokerage accounts, and network linkage fairly commonplace.

8.4

Futures Contract Features and Price Quotes

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This section discusses a futures contract’s specifications and how prices and trades are recorded.

Commodity and Financial Futures Contracts Futures contracts are divided into those on commodities or financial futures. Commodity futures usually refer to futures contracts on commodities like cattle, coffee, copper, and corn. Financial futures refer to futures contracts on financial assets like bonds, currencies, or even on notional variables like stock indexes and interest rates (see Table 8.4).1 Greater connectivity among exchanges has considerably expanded the menu of choices available for trading futures. Most futures have similar features, especially those trading on US exchanges. As “seeing one is seeing them all,” we select a popular contract and examine its specifications. Gold is the natural choice since gold futures trade on several US exchanges and on at least ten foreign ones.

The Gold Futures Contract Let us examine some features of the gold futures contract that trades on the COMEX division of the CME Group (see Example 8.1).2 1

Legally the term commodity has a broader definition that includes nearly all goods and services (see Title 7 [Agriculture], Chapter 1 [Commodity Exchanges], Section 1a of the US Code), and under this legal definition, financial futures are considered commodity futures. 2

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EXAMPLE 8.1: The Gold Futures Contract Traded on the COMEX Division of the CME Group ■

A trading or ticker symbol identifies the futures contract. The gold futures contract has the ticker symbol GC.

■

A trading month identifies the contract’s delivery month. At any time, GC contracts trade with “delivery during the current calendar month; the next two calendar months; any February, April, August, and October falling within a 23-month period; and any June and December falling within a 72-month period beginning with the current month.”3 Table 8.5 gives actual price quotes for these contracts and shows some months that trade. The choice of trading months is crucial for many commodities. As gold is produced steadily throughout the year, it makes sense to have delivery months spaced at regular intervals throughout the year. Futures on agricultural commodities often have delivery months clustered around harvest times.

■

The trading unit or contract size (also called even lot) is one hundred troy ounces per contract. Some commodities including gold have futures contracts of multiple sizes.

■

Trading hours In both the CME Globex (electronic trading system) and CME ClearPort (system for clearing over-the-counter trades), trading takes place from 24 hours a day (with a daily break from 4 PM to 5 PM Chicago time), seven days a week. Floor trading for gold futures has been phased out.

■

The price quotation is in US dollars and cents per troy ounce. If the quoted price is $500, then the position size for one contract is 500 × 100 = $50,000.

■

The minimum price fluctuation or tick size measures the minimum price jump. GC has a tick size of 10 cents ($0.10) per troy ounce or $10 per contract. For example, if the gold futures price is $500.00, the next higher price would be $500.10, the next would be $500.20, and so on. The long position gains and the short position loses by $10 for each of these jumps. If the price falls from $500.00 to $498.50, then the long loses $150 per contract.

■

The maximum daily price fluctuation or daily price limit is the maximum price change allowed in a day. Some commodities have price limits (see the relevant exchange websites for details).

■

The last trading day is the third to last business day of the delivery month. This is the last day on which a GC contract maturing in that month can trade.

■

Delivery is the traditional way of ending a futures contract. Gold delivered against a GC contract must bear a serial number and an identifying stamp of a refiner approved and listed by the exchange. Delivery must be made from a depository that has been licensed by the exchange.

■

Besides delivery, a contract may end with an exchange of physicals (EFP) for the futures contract. In this case, the buyer and the seller privately negotiate the exchange of a futures position for a physical position of equal quantity. EFPs take place at locations that are away from the exchange floor.

■

The delivery period is the time period during which delivery can occur. The first delivery day is the first business day of the delivery month, while the last delivery day is the last such day. Like most futures, the GC seller decides if a delivery should take place, and if so, when to start the process and what grade to deliver. This makes sense because historically, sellers were hedgers who held the physical commodity for sale.

FUTURES CONTRACT FEATURES AND PRICE QUOTES

TABLE 8.4: Examples of Futures Contracts That Trade in the United States Category

Examples of Commodities

COMMODITY FUTURES Grains and oilseeds

Corn, soybean, wheat, canola, . . .

Livestock and meat

Cattle, pork belly, hogs, . . .

Food and fiber

Coffee, orange juice, sugar, cotton, . . .

Metals

Copper, gold, platinum, silver, . . .

Energy

Gasoline, heating oil, natural gas, electricity, . . .

Other products

Ammonia, lumber, plywood, . . .

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FINANCIAL FUTURES Interest rates

Canadian bonds, euribor, eurodollars, Treasury securities, . . .

Currencies

Euro, peso, pound, Swiss franc, . . .

Indexes

Dow, Nasdaq, Nikkei, S&P 500, . . .

■

Trading at Settlement allows traders to trade any time during trading hours but at a delivery price that is the settlement price determined at the day’s close. This is only allowed for certain months.

■

Regarding grade and quality specifications, in case of delivery, the seller must deliver 100 troy ounces of gold, subject to various specifications regarding assaying fineness, the form in which it can be delivered, and acceptable refiners (details are available from the exchange on request.)

■

The position limit restricts the number of contracts that a speculator can hold in particular commodities (see the relevant exchange websites for details). Position limits try to prevent traders from manipulating by accumulating huge long positions and then squeezing short sellers. Several forward and futures market manipulation cases discussed in chapter 10 show that this risk is real.

■

Margins are security deposits required to open and maintain futures positions. A nonmember opening a speculative position has to keep an initial margin (security deposit needed to open a futures position) per contract. The maintenance margin is the amount that if the account value falls to this level, the trader has to supply more funds to bring the margin back to the initial margin. For the exact dollar margins required, see the relevant exchange websites.

3

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TABLE 8.5: COMEX Gold Futures Prices at Market Close on Thursday, August 24, 2007

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GOLD (Comex Division, NYMEX) 100 troy oz.; $ per troy oz.

Last

Open High

Open Low

High

Low

Aug 2007

668.10

n/a

662.10

668.80

662.10

Sep 2007

667.00

659.60

659.60

667.00

659.60

Oct 2007

671.20

665.00

664.00

673.10

661.00

Nov 2007

n/a

n/a

n/a

n/a

n/a

Dec 2007

678.00

671.20

670.90

679.00

666.70

Feb 2008

683.00

n/a

673.90

683.10

673.00

April 2008

679.40

679.40

679.40

679.40

679.40

June 2008

694.00

685.00

685.00

694.00

685.00

Aug 2008

n/a

n/a

n/a

n/a

n/a

Oct 2008

n/a

n/a

n/a

n/a

n/a

Dec 2008

710.00

n/a

701.30

710.00

701.30

Feb 2009

n/a

n/a

n/a

n/a

n/a

April 2009

n/a

n/a

n/a

n/a

n/a

June 2009

n/a

n/a

n/a

n/a

n/a

Dec 2009

n/a

n/a

n/a

n/a

n/a

June 2010

n/a

n/a

n/a

n/a

n/a

Dec 2010

n/a

n/a

n/a

n/a

n/a

June 2011

n/a

n/a

n/a

n/a

n/a

Dec 2011

n/a

n/a

n/a

n/a

n/a

June 2012

n/a

n/a

n/a

n/a

n/a

Gold Futures Price Quotes Wire news services, websites, and many major daily newspapers carry futures price quotes. As there are some differences in how they report, we discuss price quotes for the COMEX gold futures. Table 8.5 gives these prices for Thursday, August 24, 2007, from the NYMEX website (this contract currently trades on the CME Group’s COMEX division; quotes are available online4 ). Newspapers provide basic

FUTURES CONTRACT FEATURES AND PRICE QUOTES

165

8.6: COMEX Gold Futures Prices at Market Close on Thursday, August 24, 2007

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mex Division, NYMEX) 100 troy oz.; $ per troy oz.

Most Recent Settle

Change

Open Interest

Estimated Volume

Last Updated

668.00s

+0.10

261

204

8/24/2007 2:57:39 pm

668.70s

−1.70

234

41

8/24/2007 2:58:42 pm

671.50s

−0.30

35815

2069

8/24/2007 3:32:29 pm

n/a

0

n/a

n/a

n/a

677.50s

+0.50

188352

59490

8/24/2007 3:41:59 pm

683.40s

−0.40

16264

659

8/24/2007 1:49:42 pm

689.10s

−9.70

17710

9

8/24/2007 1:41:15 pm

694.70s

−0.70

14606

37

8/24/2007 1:42:53 pm

700.10s

0

7541

47

8/24/2007 1:41:15 pm

705.50s

0

1469

n/a

8/24/2007 1:41:15 pm

710.90s

−0.90

15978

181

8/24/2007 1:41:15 pm

716.60s

0

11076

n/a

8/24/2007 1:41:15 pm

722.20s

0

1630

n/a

8/24/2007 1:41:15 pm

727.80s

0

10983

n/a

8/24/2007 1:41:15 pm

744.90s

0

2566

n/a

8/24/2007 1:41:15 pm

762.30s

0

2569

n/a

8/24/2007 1:41:15 pm

780.10s

0

2585

n/a

8/24/2007 1:41:15 pm

798.10s

0

1724

n/a

8/24/2007 1:41:15 pm

816.50s

0

1035

n/a

8/24/2007 1:41:15 pm

835.90s

0

0

n/a

8/24/2007 1:41:15 pm

information like where the contract trades, the size of the contract, and how prices are quoted. This redundant information is not reported on the NYMEX website. The first column lists the delivery months. The last trade price is reported next: this was $678.00 for the December 2007 contract. Many exchanges begin each day’s trading with an opening call period for each contract month. The first few bids, offers, and traded prices during this initial time period establish an opening price or an opening range. This can give rise to different open high and open low prices,

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which were $671.20 and $670.90, respectively, for December 2007. Regular trading begins after everyone gets a chance to execute trades at the opening call. High and low are the highest and the lowest traded prices per contract per day. The high price for December 2007 gold was $679.00, while the low was $666.70. The difference between the high and the low prices determines the trading range for the trading session. Like the opening, trading ends in a special way. The closing minutes are usually the busiest because this is when many traders close out their open trades to avoid margins for overnight positions. During the brief ending period, called the market close, or the last minute of trading, known as the closing call, a closing price or a closing range is established. For actively traded contracts with few price fluctuations, the exchange’s Settlement Committee picks a settlement price from this closing range, often the last traded price. The settlement price (abbreviated settle) is the fair value of the contract at market close. For December 2007, the most recent settlement price was $677.50. The task is harder for thinly traded contracts. For example, April 2008 traded nine contracts, which may have traded hours before the close. Moreover, many contracts, like February, April, and June for the year 2009, did not trade at all. In this case, the Settlement Committee considers the spreads relative to other futures prices. The spreads between different maturity futures prices are generally quite stable. For example, the spread between the April and February 2008 settlement prices was $5.70, between June and April $5.60, and between August and June $5.40. Here the spreads between the February and April 2009 and the April and June 2009 settlement prices (contracts with no current trading volume) were set at $5.60. The next column reports the change in the settlement price from the day before. For example, the December 2007 settlement price rose by $0.50 to end at $677.50. Some tables report lifetime highs and lows, which refer to the highest and the lowest traded prices recorded for a contract month since it began trading. This information is absent here. Open interest is the number of outstanding contracts for a particular maturity month. December 2007 has 188,352 open contracts. Each of these contracts must end through a closing trade, a delivery, or a physical exchange. The estimated trading volume during the current session is 59,490 contracts for December 2007, and the last updated gives the time of the last trade.

The Exchange and Clearinghouse US securities exchanges are governed by a tight set of rules and regulations. Traditionally, they have been organized as voluntary, nonprofit institutions. Futures exchanges in the United States have clearinghouses affiliated with them. A clearinghouse forms the foundation on which a futures exchange is built. As noted before, it serves several useful functions: it clears trade (and collects a clearance fee for each contract cleared), it guarantees contract performance to all traders, and it collects margin from

COMMODITY PRICE INDEXES

clearing members. A clearinghouse has matched books; therefore it has no market risk, but it does have counterparty risk. To alleviate this credit risk, it has several layers of financial protection. On top of margins, a member must keep guarantee funds with the clearinghouse. In case a member defaults, the clearinghouse first pays from his margin, then from the member’s guarantee fund, subsequently from the surplus fund of clearance fees, and finally from the guarantee fund of all members according to some prespecified formula.

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8.5

Commodity Price Indexes

We have used equity market indexes, the Dow Jones Industrial Average, and the S&P 500 index to understand the history of stock markets, to gauge market sentiments, to evaluate portfolio performance, and even as the underlying for derivatives. Can indexes also play such roles in commodity markets? The answer is yes. During the 1950s and 1960s, inflation was low, commodity prices were relatively stable, and volatility was not a matter of concern. The 1970s, 1980s, and 1990s exhibited more volatility but no discernable trend in the index. A dramatic upward trend began in 2003 with increased volatility. Not surprisingly, this increased volatility has been accompanied by an increase in the trading of commodity futures. Extension 8.1 presents an interesting debate concerning the direction of future commodity prices between the doomsters and the boomsters. Several commodity price indexes have been created. The most popular commodity price index, the Reuters/Jeffries CRB index (RJ/CRB), is computed by averaging the futures prices of nineteen different commodities. Several other commodity price indexes are popular in the United States. Standard and Poor’s publishes world production–weighted S&P GSCI indexes based on twenty-four commodities, which consist of four separate but related S&P GSCI indexes: the spot index, the excess return index, the total return index, and a futures price index. Introduced in 1998, the Dow Jones–AIG Commodity Index Family (DJ-AIGCI) is composed of indexes based on futures contracts on nineteen physical commodities. DJ-AIGCI indexes are computed on an excess and total return basis, report spot as well as forward indexes, and are available in US dollar and several foreign currency versions. The Rogers International Commodity index (RICI) was developed in 1998 by commodity investor James B. Rogers (see RICI Handbook 2008). Covering more commodities than other indexes, RICI is a dollar-based total return index that corresponds to a collection of commodities representing the global economy including futures traded in different exchanges, in different countries, and quoted in different currencies. Notice the following about these indexes. First, as these indexes are weighted averages, the booming energy sector gets high weights, ranging from 30 plus percent for RJ/CRB and DJ-AIGCI to over 70 percent for the S&P GSCI index in 2008. Second, because these indexes are based on short-lived futures contracts that mature on a regular basis, the managers of the index must periodically roll the futures index

167

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by replacing the expiring futures with similar new contracts. Third, many derivatives have been created based on these indexes. For example, there are exchange-traded funds or exchange-traded notes based on these indexes, and futures or options on futures trade on these indexes. Exchange-traded funds (ETFs) are similar to mutual funds but trade in an exchange on a real-time basis. Exchange-traded notes are bonds issued by an underwriting bank, which promises to pay a return (minus any fees) based on the performance of a market benchmark (like a commodity price index) or some investment strategy.

EXTENSION 8.1: Doomsters and Boomsters

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Two Opposing Views on the Future Direction of Commodity Prices The first camp, popularly known as doomsters, owes its intellectual origins to the great English economist Thomas Malthus, whose ideas on the economics of population growth came to be known as the Malthusian doctrine. It argues that as productivity increases and new lands are discovered, the food supply increases in arithmetic proportions, such as 1, 2, 3, . . . . However, due to the population’s propensity to reproduce, it increases geometrically, such as 1, 2, 4, 8, . . . . Consequently, events like war, famine, and pestilence check the population growth relative to the food supply so that people always stay just at the subsistence level. No wonder economics has been called the dismal science! Building on this intellectual foundation, the doomster school argues that a growing number of humans rely on a limited supply of natural resources, and scarcity will force commodity prices to rise through time. A prominent proponent of this neo-Malthusian view is Stanford University professor Paul R. Ehrlich. His 1968 book The Population Bomb warned, “The battle to feed all of humanity is over. In the 1970s and 1980s hundreds of millions of people will starve to death in spite of any crash programs.” The opposite camp, popularly referred to as boomsters, espouses free-market environmentalism, which argues that free markets, property rights, and a good legal system will take care of the environment. They dismiss the other side’s gloom and doom. They cite that after the high-inflation days of the 1970s, commodity prices failed to show a pronounced upward trend implied by increasing scarcity. The champion of the second group was Julian L. Simon (b. 1932), a professor at the University of Maryland when he died in 1998. His 1981 book The Ultimate Resource challenged the notion of Malthusian catastrophe and offered an alternate explanation. He argued that the power of human beings to invent and adapt is the ultimate resource. Simon showed that after adjusting for inflation and wage increases, most raw material prices fell over the past decades. To quote from a song by the Beatles, “It’s getting better all the time.”

A Bet between Professor Doomster and Professor Boomster Boomster Simon and doomster Ehrlich entered into a wager based on their views (see “When the Boomster Slams the Doomster, Bet on a New Wager—Clash of Eco-Titans Rekindles a Rancorous Core Debate over Future of the Earth,” Wall Street Journal, June 5, 1995). The bet’s conditions were as follows: on September 29, 1980, Ehrlich would buy on paper $200 worth of each of five strategically important metals (chromium, copper, nickel, tin, and tungsten). If, ten years later, their inflation-adjusted prices went up, then Simon would pay the inflation-adjusted value increase for each of these portfolios. If they declined, Simon would receive a payment instead. Note that the professors were trading a derivative that was essentially a combination of five forward contracts with some price adjustments.

SUMMARY

169

Ehrlich and his colleagues saw an arbitrage opportunity: “I and my colleagues, John Holdren (University of California, Berkeley) and John Harte (Lawrence Berkeley Laboratory), jointly accept Simon’s astonishing offer before other greedy people jump in.” Ten years later, they were in for a rude surprise—for a variety of reasons, the real prices for all these metals and the nominal prices for three declined. Wired magazine (“The Doomslayer,” www.wired.com/wired/archive/5.02/ffsimon_pr.html) reports that the drubbing was particularly hurtful because Simon had given Ehrlich and his colleagues a priori advantage by letting them select the five metals. A month after the bet ended, Professor Ehrlich quietly mailed Professor Simon a check for $576.07.

8.6

Summary

1. Both a forward and a futures contract fix a price for a later transaction. They have many differences: unlike a forward, a futures contract is (1) regulated, (2) exchange traded, (3) standardized, (4) liquid, (5) guaranteed by a clearinghouse, (6) margin adjusted, (7) daily settled, (8) usually closed out before maturity, and (9) has a range of delivery dates.

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2. Futures and forwards have many uses: (1) they help smooth out price fluctuations that can come from demand–supply mismatches, (2) they help create a complete market, (3) they help people to speculate, (4) they allow traders to leverage their capital, (5) they make efficient trading possible, and (6) they help generate information and aid in price discovery. 3. Forward trading is as old as antiquity. Today’s futures contract started in nineteenthcentury Chicago and more or less arrived at its present-day form over the next eighty years. An astonishing variety of futures were introduced since the 1970s— financial futures being a major innovation. Moreover, exchanges merged, new exchanges emerged, and electronic trading replaced the traditional open-outcry method of pit trading throughout the world. 4. As futures are standardized, they allow quicker transactions. Futures contract specifications are available on an exchange’s website. 5. Many vendors carry futures price quotes. A typical quote table gives the opening price, the day’s high and low, the settlement price, the change in settlement price from the previous day, lifetime high and low, open interest for each contract, and other information like the estimated volume and open interest for all contracts on that particular commodity. 6. Several commodity price indexes like the Reuters/Jeffries CRB index, S&P GSCI indexes, Dow Jones–AIG Commodity Index Family, and Rogers International Commodity index are used to gauge the direction of commodity prices and to create derivatives that allow investors to add a commodity price exposure to their portfolios.

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8.7

Cases

CME Group (Harvard Business School Case 711005-PDF-ENG). The case describes

the CME Group, the world’s largest commodities exchange, futures and options on futures contracts, history, regulation, and the strategic choices the company faced. Bringing OTC Back to the Exchange: Euronext. liffe’s Launch of ABC

(Harvard Business School Case 706489-PDF-ENG). The case examines value creation, market design, and competitive positioning issues for a derivatives trading exchange in the context of launching matching, clearing, and confirmation services for the over-the-counter market. Lessons Learned? Brooksley Born and the OTC Derivatives Market (A)

(Harvard Business School Case 311044-PDF-ENG). The case studies a proposal to regulate the over-the-counter derivatives market, whose lack of implementation might have been one of the factors that contributed to the financial crisis of 2007–9.

8.8

Questions and Problems

8.1. Where and when was the world’s first modern futures exchange founded?

Explain how trading took place on this exchange. 8.2. Discuss the four different categories into which the history of futures contracts

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since 1970 has been classified, giving two examples of some of the major developments under each of the four categories. 8.3. Consider a fairly illiquid futures contract that has not traded for days. Do you

still need a settlement price for this contract? If so, how would the exchange go about determining this settlement price? 8.4. Explain the difference between closing price and settlement price. Are they

the same or different? Which price is used for marking-to-market? 8.5. If regulation is bad for business, why does the National Futures Association put

a significant amount of regulation on its members? 8.6. Consider the gold futures contract traded in the COMEX division of the CME

group. What is the trading unit size and minimum tick size? 8.7. Using Table 8.5, what is the last price on the June 2008 futures contract? What

does the last price mean? 8.8. Using Table 8.5, what is the open interest on the June 2008 futures contract?

What does the open interest mean? 8.9. Using Table 8.5, looking at the open, high, and low prices on the June futures

contract, what can you tell about the trend of futures prices on this contract during the day? 8.10. Using Table 8.5, which three futures contracts have the most trading activity?

QUESTIONS AND PROBLEMS

8.11. Today, are futures contracts only traded on US exchanges? Explain your answer. 8.12. Is trading in futures contracts mainly for hedging, or speculation, or both?

Explain your answer. 8.13. Explain why forward contracts have been trading for centuries. What economic

function do they perform? What improvement did futures contracts provide over forward contracts? Explain your answer. 8.14. The following questions concern the doomsters and the boomsters. a. What were the opposing views promoted by Professors Ehrlich and Simon? b. Describe the wager between Professors Doomster and Boomster and the

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outcome of this bet.

171

9 Futures Trading 9.1 Introduction 9.2 Brokers, Dealers, and the Futures Industry

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9.3 Trading Futures 9.4 Margin Accounts and Daily Settlement Daily Settlement

9.5 Futures and Forward Price Relations

Equality of Futures and Spot Price at Maturity Equality of Forward and Futures Prices before Maturity Convergence of the Basis

9.6 Trading Spreads 9.7 Summary 9.8 Cases 9.9 Questions and Problems

BROKERS, DEALERS, AND THE FUTURES INDUSTRY

9.1

Introduction

The movie Trading Places made a caricature of futures trading. In a particular sequence, Dan Aykroyd and Eddie Murphy entered the trading pit, visibly nervous and yet pumped up to speculate by shorting orange juice futures. They suddenly realized that the powerful Duke brothers were trying to corner the OJ futures market. This was in anticipation of crop damage from a frost in the orange-growing states. Some traders thought that the Duke brothers had inside information on the extent of the losses and joined the fray. Futures prices increased. Dan, Eddie, and other sellers soon felt that they were sliding down a bottomless pit. Suddenly it all changed. The commerce secretary announced on television that the frost had bypassed the orangegrowing regions. This implied a rich harvest with soft prices. Dan and Eddie were ecstatic: OJ futures prices rapidly fell, and they made a ton of cash; the Duke brothers went completely broke—and all of this happened in an hour! Although the movie is fictionalized, it does capture the dark side of futures trading (it is suspiciously similar to the Hunts brothers’ silver manipulation story in the next chapter). To put this movie in context, this chapter describes the mechanics of trading futures contracts. We discuss brokers, dealers, and others who work in the industry, then we list various ways of moving in and out of futures. This is followed by a description of margins and daily settlement. Next we explore various properties of futures and forward prices. We end with a discussion of futures spread trading strategies.

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9.2

Brokers, Dealers, and the Futures Industry

Brokers match buyers and sellers in futures and earn commissions for this service. They also represent their customers to the exchanges and clearinghouse. Dealers step in and take the other side of trade when no one else offers a better price. Like a used car dealer who maintains an inventory and posts a retail price for trading cars, dealers in futures markets keep an inventory of futures contracts and quote bid and ask prices to maximize income. Moreover, brokers and dealers also act as first-level regulators because they have a duty to report irregularities and fraud to the exchange authorities and government regulators. Brokers and dealers in futures markets may do business as individuals, or they may be organized as associations, partnerships, corporations, or trusts. They must register with the Commodity Futures Trading Commission (CFTC). To open a futures trading account, one must go through a futures commission merchant (FCM) or an introducing broker (IB) registered with an FCM. An IB’s role and responsibilities are limited: an IB may seek and accept orders but must pass them on to a carrying broker (usually an FCM) for execution, and an IB cannot accept funds—funds must be directly deposited with the carrying broker. In contrast, an FCM provides a one-stop service for all aspects of futures trading: solicit trades, take futures orders, accept payments from customers, extend credit to clients, hold margin deposits, document trades, and maintain accounts and trading records. Numerous Associated Persons (APs)

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work for FCMs, IBs, and others, and help people trade (see CFTC’s website https://www.cftc.gov/ConsumerProtection/EducationCenter/registration.html for a description of the intermediaries who facilitate the trading process). Where does one go for trading advice? There are commodity trading advisors who dispense wisdom through newsletters or advise clients on an individual basis. Most of them are technical analysts who use past price patterns to predict future price movements. To invest with professional managers, one can go to commodity pool operators (CPOs). They run commodity pools (funds), which are mutual fund–type operations speculating in futures. Rarely are these pools stellar performers: the returns are often negative, and when positive, they often underperform the market on a risk-adjusted basis. Moreover, they charge large management fees and require big margin deposits.

9.3

Trading Futures

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Entering into a long or a futures position is easy: one calls a broker and places an opening order. But, there are four ways of ending a futures position: a closing transaction, physical delivery, cash settlement, or an exchange for physicals (see Figure 9.1): 1. A closing transaction. Suppose you go long a December gold futures. You are obliged to buy gold at the futures price during the December delivery period. A futures position can be ended earlier. You can do this by an offsetting transaction called a closing transaction or a reversing trade. Just take a short position in the same December gold futures. You are now obliged to sell gold at that day’s futures price during December. This cancels the initial position. Tell your broker that the second trade was a closing transaction, and he will wipe your slate clean. Over 95 percent of futures contracts close before maturity. 2. Physical delivery. This was the only way of closing futures in the past. In a physical delivery, the short gets cash from the long and obtains a warehouse receipt that gives ownership of goods held in a licensed warehouse or a shipping certificate that is a promise by an exchange-approved facility to deliver the commodity under specified terms. A futures contract specifies (1) acceptable deliverable grades (some contracts allow substitution of inferior grades with price discounts and superior grades with price premiums), (2) acceptable delivery dates (usually within the last trading month, i.e., the month that includes the last trading day), and (3) acceptable delivery places (with price adjustments for different locations). Though conceptually similar, the delivery methods vary slightly from exchange to exchange. For most contracts, the short has the privilege of initiating a delivery notice. The exchange usually matches him with the oldest existing long position. In a typical delivery process, there would be a first notice day, a last notice day, and a last trading day. The first notice day is the first day on which a short can submit a notice of intention to make delivery, and the last notice day is the last such day. The last trading day is usually one or more days before the last notice day. Exchanges generally do not make or take delivery of the actual commodity; they only specify how the delivery process occurs.

TRADING FUTURES

FIGURE 9.1: Four Methods of Ending a Futures Position Closing Futures before the Delivery Period Start

Delivery Period

Last Trading Day (a) Closing transaction (preferred way of ending most futures contracts) Ending Futures during the Delivery Period Corn moves Colorado Farmer Corn Processor

Chicago

Kansas City

(b) Physical delivery (inconvenient, large transaction costs, e.g., farmer ships corn to Chicago or Kansas City and delivers corn for cash; processor has to transport corn back to Colorado) (c) Cash settlement (no need to ship the underlying commodity; the contract ends with a final margin adjustment on the last trading day)

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Traders inform exchange

Chicago

Colorado Farmer Corn moves Corn Processor (d) Exchange for physicals (hedgers avoid needless transaction costs, e.g., a farmer and a corn processor arrange delivery for payment and inform the exchange to close positions)

3. Cash settlement. In case of a cash settlement, a final adjustment is made to the margin accounts of the buyer and seller equaling the difference between the last trading day’s settlement price and the settlement price of the previous day—and then the contract ends. 4. Exchange for physicals. Another possibility is a spot market trade between two parties who have genuine buying and selling needs for the underlying commodity and have already hedged their prices by establishing futures positions. This is called an exchange for physicals (EFP) transaction. Afterward, they notify the exchange and the clearinghouse about the transaction, and their slates are rubbed clean. Example 9.1 shows how this works.

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EXAMPLE 9.1: An Exchange for Physicals ■

Suppose a farmer grows corn in Colorado. She sells short fifty corn futures to fix a selling price for her produce. A corn processing plant is located five miles up the road from the farmer. It buys, processes, and sells corn to the cereal maker who makes cornflakes. The corn processor goes long fifty corn futures and fixes a buying price for the input. Both traders are hedgers: the farmer genuinely needs to sell corn, while the processing plant legitimately needs to buy it.

■

At delivery time, why ship 250,000 (= 50 contracts × 5,000 bushels/contract) bushels of corn all the way to Chicago or Kansas City and then ship it back to Colorado as per CME Group’s contract stipulations? This will incur needless transportation costs and time delays.

■

It would be much better to do the transaction directly: the farmer ships corn to the processing plant and receives the day’s settlement price (although provisions allow EFPs at a mutually agreed price). Both report the price and the quantity traded to the Chicago Board of Trade, which then checks that these reports match and closes the fifty long and short positions.

EFPs have grown in popularity in recent years. Most oil futures are settled by EFPs, and their use is increasing in both gold and silver futures markets.

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9.4

Margin Accounts and Daily Settlement

To buy and sell futures, one must open a futures account with a broker. To open a futures position, one must post initial margin (or performance margin), which acts as a performance bond. Most exchanges allow initial margins to be paid with cash, bank letters of credit, or short-term Treasuries. The broker, if she is a clearing member, keeps margin deposits with the exchange’s clearinghouse; if she is not, she keeps them with a clearing member, who in turn keeps margin with the exchange. This earnest money ensures that when the going gets tough, the tough do not leave town!

Daily Settlement When futures prices increase, a long position gains value. This makes sense: if you buy something at a fixed price and the new purchase price increases, then you must make a profit. The seller’s situation is just the opposite. So when the futures price increases, the long’s account is credited with variation margin, which is the change in the settlement price from the previous trading day, and the short’s account is debited the same. These additions (collects) or subtractions ( pays) of variation margins at the end of each trading day are known as daily settlement. When this is done, the margin account is said to be marked-to-market. Traders can keep only a small amount as margin, often 5 percent or less of the position size (e.g., if the gold futures price is $1,000 and the initial margin is $5,000, then it is only 5,000/(1,000 × 100) = 0.05 or 5 percent), and still trade.

MARGIN ACCOUNTS AND DAILY SETTLEMENT

Marking-to-market substantially lowers credit risk and makes futures safer to use than forward contracts. The reason is simple. With futures, the largest loss to the contract is a one-day movement in the futures price. In contrast, with forwards, the largest loss is the movement in the forward price over the entire life of the contract. Margins buffer this smaller loss for futures, whereas collateral buffers this larger loss for forwards. Example 9.2 shows how daily settlement works for a futures contract.

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EXAMPLE 9.2: Daily Settlement of a Gold Futures Contract ■

On Monday, Ms. Longina Long buys three gold futures contracts from Mr. Shorty Short. Both traders have margin accounts. We refer to Monday as time 0, Tuesday as time 1, and so on. The futures contract matures on Friday (time T or 4) when the day’s trading ends. For convenience, we use the end of the day’s spot and settlement prices. All prices and computations are on a per ounce basis. The futures prices are obtained from the cash-and-carry model (see Chapter 11).

■

Monday (time 0) - The spot price on Monday, S(0), is $1,000.00. - The futures price on Monday for the contract maturing on Friday is F(0,4) = $1,000.60. For simplicity, we suppress T from the notation for the futures price F(0,T) and write it as F(0). We will reintroduce it later when considering contracts maturing at different dates. Table 9.1 records the trading day as the first column and futures prices as the second column. - Because futures contracts always clear at fair prices, no cash changes hands when the position is entered at the close of Monday’s trading.

■

Tuesday (time 1) - S(1) = $1,006.00 - F(1) = $1,006.40 - Yesterday, Ms. Long locked in a price of $1,000.60 for buying gold on Friday. Now she is asked to pay $1,006.40 instead. She is understandably upset. However, she will be indifferent if Mr. Short pays her the price difference. By daily settlement, this amount, F(1) – F(0) = 1,006.40 – 1,000.60 = $5.80, is credited to Long’s margin account (and it is debited to Short’s margin account) after Tuesday’s close. This is reported in the third column of Table 9.1. Long’s contract now has a futures price of F(1) = $1,006.40.

■

Wednesday (time 2) - S(2) = $996.00 - F(2) = $996.20 - Long would be happy to buy gold for $996.20 on Friday, while Short would be unhappy to sell at that price. To mark-to-market the contract, Long’s account is debited by a variation margin of F(2) – F(1) = 996.20 – 1,006.40 = – $10.20. Short’s account is credited the same amount. Values become fair again.

■

Thursday (time 3) - S(3) = $988.00 - F(3) = $988.20 - Long’s cash flow is F(3) – F(2) = $988.20 – 996.20 = – $8.

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Friday (time T or time 4) - S(4) = $995.00 = F(4) - The spot and futures prices equal each other at the end of Friday’s trading. They should because a futures contract maturing in an instant is the same as a spot market transaction and therefore the futures price must be the same as the spot price. - Long gets F(4) – F(3) = 995.00 – 988.20 = $6.80 from Short.

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TABLE 9.1: Futures Prices and Margin Account for Longina Long Day

Futures Settlement Price

Change in Futures Settlement Price

Daily Futures Gain or Loss (Last Column × 300)

Mon

F(0) = $1,000.60

Tues

F(1) = 1,006.40

5.80

1,740

Wed

F(2) = 996.20

–10.20

Thurs

F(3) = 988.20

Fri

F(4) = 995.00

Interest Earned on Previous Balance

Margin Account Balance

Margin Call

Margin Account Balance (After Adjustment)

$15,000.00

$15,000.00

1.50

16,741.50

16,741.50

–3,060

1.84

13,683.34

13,683.34

–8.00

–2,400

1.64

11,284.98

6.80

2,040

1.50

17,041.50

3,715.02

15,000.00 17,041.50

How are these adjustments actually made? If your futures position gains value, you may remove funds that are in excess of the initial margin, but if it loses value, the broker gets uncomfortable. So there is a mandatory amount called a maintenance margin (often set at 75 percent of the initial margin) that must be maintained in the account at all times. If the account value touches or drops below this level, then the broker places a margin call and requests that you come up with enough variation margin in cash to bring your account to the initial margin level. If you fail to meet a margin call, your broker can close out your positions. You may be liable for more cash if your position is liquidated at a loss. To avoid this hassle, traders often keep more funds than the required initial margin. In reality, other issues must also be reckoned with. For example, there are transaction costs, and the margin account earns interest. As this discussion can get complicated quickly, to simplify the presentation we embrace the old maxim “follow the money!” Following this maxim, we take an accountant’s perspective and keep track of the margin account’s cash flows, including daily interest (see Example 9.3).

MARGIN ACCOUNTS AND DAILY SETTLEMENT

EXAMPLE 9.3: A Margin Account’s Cash Flows Including Interest Earned ■

Continuing with the previous example, let us track the margin account of Ms. Longina Long, who buys three gold futures contracts on Monday at a futures price of F(0) = $1,000.60 per ounce. We write the number of contracts as n = 3, the contract size as 𝜅 = 100 ounces, and the margin account’s balance as Bal(t), where t = 0, 1, 2, 3, 4 are the days over which the futures contract lasts.

■

As a speculator who is not an exchange member, Long must keep the commodity exchange mandated initial margin of $5,000 per contract or $15,000 in total, which we write as Bal(0). Long’s margin account balances are plotted in Figure 9.2. Her maintenance margin is $4,000 per contract. The margin account balance earns interest. Assume that the interest rate i(0) is 1 basis point for Monday. This overnight interest rate changes randomly across time and gets declared at the start of each trading day.

■

On Tuesday, the gold futures settlement price goes up to F(1) = 1,006.40. Long’s margin account gets credited for the increase in the position’s value and earns interest on the previous day’s balance, values reported in the fourth and fifth columns of Table 9.1, respectively. Tuesday’s margin account balance is

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Bal(1) = [Futures (Tues) − Futures (Mon)] × Number of contracts × Contract size + (1 + Daily interest) × Margin account balance (Mon) = [F (1) − F (0)] × n × 𝜅 + [1 + i (0)] × Bal (0) = 5.80 × 3 × 100 + (1 + 0.0001) × 15, 000 = 1, 740 + 15, 000 + 1.50 = $16, 741.50

■

■

This amount is noted in the sixth column. As there is no margin call, we repeat this in the eighth column. Long can remove the excess margin of $1,741.50 if she wants. We assume that she keeps it in the margin account. On Wednesday, the gold futures price falls to F(2) = $996.20 and Long’s futures position loses value. Assuming Tuesday’s interest rate i(1) is 0.00011 for the day, the new margin balance is Bal (2) = [F (2) − F (1)] × n × 𝜅 + [1 + i (1)] Bal(1) = −10.20 × 300 + 1.00011 × 16, 741.50 = $13, 683.34

■

Her account is now under-margined as it has fallen below the initial margin level. The broker is uncomfortable but does nothing.

■

On Thursday, the gold futures price falls by another $8. Assuming i(2) is 0.00012 for the day, the new margin balance is Bal(3) = $11,284.98. As this is below the maintenance margin level of 3 × 4,000 = $12,000, the broker issues a margin call instructing her to come up with $3,715.02 in cash (which is reported in the sixth column of Table 9.1 and also shown in Figure 9.2). She does this and her

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account balance is restored to the initial margin account level of $15,000, and this figure is reported in the last column. Notice that futures traders are required to bring the margin account balance up to the initial margin level and not to the maintenance margin level as required in stock markets. ■

On Friday, the gold futures price goes up by $6.80. If i(3) = 0.0001, her margin account balance is $17,041.50.

■

Suppose Mr. Shorty Short took the other side of the transaction. His daily futures gains and losses would be the opposite of Long’s, but his margin account balance would earn different interest. Also, as the futures price did not move too much against Short, he will not get any margin calls.

■

As noted in Table 9.1, for the magnitude of these investments, the interest earned is small ($1.50 to $1.84) and perhaps unimportant. However, consider a futures position with a notational in the tens of millions of dollars. In that case, the interest earned per day is considerable and not so easily ignored.

FIGURE 9.2: Long’s Margin Account Balance Margin Account Balance $17,000

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16,000 Initial Margin

15,000 14,000

Margin Call Maintenance Margin

13,000 12,000 11,000 Mon

9.5

Tue

Wed

Thu

Fri

Days

Futures and Forward Price Relations

What is the relation between futures and forward prices? Before addressing that question, let’s consider an example that computes a futures contract’s payoffs when positions may be entered and exited early (see Example 9.4).

FUTURES AND FORWARD PRICE RELATIONS

EXAMPLE 9.4: Futures Payoffs When Positions Are Closed Out Early ■

Consider the data from Example 9.3. Ms. Long bought gold futures on Monday. Her payoff on Tuesday due to daily settlement is Futures (Tues) − Futures (Mon) + One day’s interest on Monday’s margin account balance = F (1) − F (0) + Interest = $5.80 + Interest

■

Instead of on Monday, if Ms. Long buys futures on Tuesday and closes out her position with a reversing trade on Thursday, then her payoff is [Futures (Wed) − Futures (Tues) + One day’s interest on Tuesday’s margin account balance] + [Futures (Thurs) − Futures (Wed) + One day’s interest on Wednesday’s margin account balance] = F (3) − F (1) + Interest = −$18.20 + Interest

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on a per ounce basis. The interest earned on the margin account balance is unknown in advance and hence random.

As indicated in Example 9.4, the interest earnings are dependent on the futures price changes. Furthermore, interest rate changes affect the interest earnings on the margin account as well. In conjunction, these two effects modify the risk of holding a futures contract relative to a forward contract that has no such cash flows. Indeed, if interest rate changes are positively correlated with futures price changes, then the futures position benefits from interest rate movements because as cash flows are received, more interest is earned. In this case, the risk of a futures position is reduced slightly relative to an otherwise identical forward contract position. Conversely, if interest rate changes are negatively correlated with futures price changes, then the futures position suffers from interest rate movements relative to a forward position. Here the risk is increased. It is important to note that this interest rate risk is priced into the market clearing futures prices, driving a wedge between futures and forward prices. Recall that forward contracts have no intermediate cash flows and therefore no explicit interest earnings risk. A generalization of this example gives a useful result for computing a futures trader’s profits and losses. Let date 0 be the initiation date of a futures contract and date T be its delivery date. Suppose a trader enters a long futures position at time t1 (which can be as early as date 0) when the futures price is F(t1 ) and closes out her position at a later date t2 (which can be as late as date T) when the futures price is F(t2 ). A futures trader’s profit or loss is the sum of daily variation margins and interest earned on the margin account balance for the time period over which the futures position is held: Long’s payoff = Closing futures price − Initial futures price + Interest = F (t2 ) − F (t1 ) + Interest

(9.1a)

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and Short’s payoff = − (Closing futures price − Initial futures price) + Interest (9.1b) = − [F (t2 ) − F (t1 )] + Interest on a per ounce basis. Of course, the interest earned on the long and short positions differ. These results make sense and ease the task of computing margin account balances (see Example 9.5). Remember that futures prices are determined in the market so that a trader entering a position pays nothing at the start. If the underlying commodity increases in value, then futures prices also move up. Long benefits because the fixed price at which she agreed to buy has increased in value, and Short loses because what he agreed to sell at a fixed price has become more expensive.

EXAMPLE 9.5: Margin Payments for a Portfolio of Futures Contracts ■

Pearl is a speculator. Trading at the prices given in Table 9.2a, she went long two contracts of gold on January 20, short three platinum contracts on January 21, and long five silver contracts on January 23. She then liquidated all her futures positions on January 26. Her gains and losses are computed in Table 9.2b. Her net gain is Long gold futures payoff + Short platinum futures payoff + Long silver futures payoff = 2, 380 − 1, 785 + 4, 975 = $5, 570

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where we have ignored the interest earned on the margin account balances for simplicity.

Let us continue with Examples 9.2 and 9.3 to explore several important properties of forward and futures prices.

TABLE 9.2A: Futures Prices Trading Date

June Gold Futures, 100 troy oz/Contract (dollars/ounce)

April Platinum Futures, 50 troy oz/Contract (dollars/ounce)

March Silver Futures, 5,000 troy oz/Contract (cents/ounce)

Jan 20

$989.1

$1,149.0

$2,542.1

Jan 21

992.8

1,153.3

2,600.0

Jan 22

989.4

1,154.1

2,553.3

Jan 23

1,007.0

1,169.5

2,623.3

Jan 26

1,001.0

1,165.2

2,643.2

Jan 27

1,008.4

1,175.0

2,651.8

FUTURES AND FORWARD PRICE RELATIONS

TABLE 9.2B: Margin Account for Pearl Portfolio

Computation of Profits/Losses

Long 2 June gold futures on Jan 20

(1,001 – 989.1) per ounce × 100 troy ounce per contract × 2 contracts

$2,380

Short 3 April platinum futures on Jan 21

–(1,165.2 – 1,153.3) × 50 × 3

–1,785

Long 5 March silver on Jan 23

(2,643.2 – 2,623.3) × 5,000 × 5 ÷ 100

4,975

Net profit

Dollar Amount

5,570

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Equality of Futures and Spot Price at Maturity In Example 9.2, suppose that at Friday’s close, the futures price is $995 but the spot price is $997. Then anyone can buy the futures, acquire gold for $995, and sell it immediately in the open market for $997 to pick up an arbitrage profit of $2. The buying pressure raises the futures price, the selling pressure lowers the spot price, and the two converge. If instead the futures price is higher than the spot price, then an arbitrageur can sell futures, make a delivery, and take an arbitrage profit. To prevent arbitrage opportunities, the futures price must equal the spot price at the maturity of the contract. Of course, we need to make assumptions such as that there are no market frictions (such as transaction costs [brokerage commissions, bid/ask spread], margin requirements, short sales restrictions, taxes that may be levied at different rates, and assets that are not perfectly divisible), that spot and futures can be simultaneously traded, and that the futures mature on a single day.

Equality of Forward and Futures Prices before Maturity Are forward and futures prices equal before maturity? They are equal only under a restrictive set of assumptions such as constant interest rates, no market frictions, and no credit risks, but not otherwise. Example 9.6 explores this and related issues.

EXAMPLE 9.6: Equality of Forward and Futures Prices ■

Consider the gold futures contract introduced in Example 9.2. Next, consider a gold forward contract that is similar to this futures contract. It begins on Monday, matures on Friday, and has one hundred ounces of pure gold as the underlying. We compare two trades by Ms. Longina Long entered at Monday’s close: buying one forward contract versus buying one futures contract.

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■

First, assume an idealistic setting in which interest rates are assumed to be zero, market frictions are assumed away, and counterparties have no credit risk. Now let us compare the payoffs on a per ounce basis from forward and futures trades.

■

At Friday’s close, Ms. Long’s payoff from a long forward trade is = Spot (Fri) − Forward (Mon) = 995 − F (Mon)

(9.2a)

where F(Mon) is the forward price for the contract maturing on Friday. ■

At Friday’s close, Ms. Long’s payoff from a long futures trade is = [Futures (Tues) − Futures (Mon)] + [Futures (Wed) − Futures (Tues)] + [Futures (Thurs) − Futures (Wed)] + [Futures (Fri) − Futures (Thurs)] = 995 − F (Mon)

(9.2b)

where F(Mon) is the futures price on Monday. This happens because Futures(Fri) = Spot(Fri) = $995 at Friday’s close. ■

The two payoffs must be equal because they acquire the same asset on the same date while requiring no cash when initiated. Hence, comparing expressions (9.2a) and (9.2b), we get F (Mon) = F (Mon) = $995

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■

(9.3)

Let us allow interest rates to be random, while holding the other assumptions the same. Again, let’s compare payoffs from a forward and futures trade. - At Friday’s close, Ms. Long’s payoff from a long forward trade is = 995 − F (Mon)

(9.4a)

- At Friday’s close, Ms. Long’s payoff from a long futures trade is = 995–F (Mon) + Interest earned on margin account balance over the time period during which futures position is held

(9.4b)

on a per ounce basis. - Equate the payoffs as before and rearrange terms to get F (Mon) = F (Mon) + Interest

(9.5)

■

Expression (9.5) shows that futures and forward prices are structurally different. This is because a futures trader earns interest on her margin account balances but a forward trader does not.

■

Introducing additional market imperfections such as transaction costs serves to complicate the relation between forward and futures prices even more.

TRADING SPREADS

■

In addition, forward and futures contracts face different risks, on top of the market risks discussed earlier. Let us consider the three additional risks stated in Chapter 1 (credit, liquidity, operational) and see how each affects forward and futures contracts differently. - Being exchange traded, a futures contract is essentially free from credit risk; a forward contract may not be. It depends on the collateral relationships prearranged between the relevant counterparties. - Over-the-counter traded forwards are more illiquid than are the exchange-traded futures, introducing greater liquidity risk in forward contracts. - Being an over-the-counter instrument, a forward also has more operational risk than does an exchange-traded futures contract.

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This example demonstrates that under the assumption of zero interest rates, forward and futures prices are equal. But what happens if interest rates are nonzero? In this case, it can be shown (see Jarrow and Turnbull [2000] for a proof) that forward and futures prices are equal, but only if interest rates are nonrandom. This is an improvement over the first assumption, but it is still an unreasonable one. Interest rates are random and wiggle unpredictably across time. If one allows interest rates to be random, which is the truth, then the equality between forward and futures prices no longer holds. The primary reason is that futures earn interest on the margin account, while forward contracts do not. The equality argument that gave us expression (9.3) does not apply. This is demonstrated by continuing the previous example. The moral of the story is that forward and futures contracts are different securities and, consequently, have different prices. They are fraternal twins, not identical!

Convergence of the Basis The spot price minus the futures price is called the basis. Although it randomly changes across time, it eventually becomes zero at maturity. This well-known property of futures prices is known as convergence of the basis. We can easily see from Example 9.4 that the basis takes on the values –0.60 (Monday), –0.40 (Tuesday), –0.20 (Wednesday), –0.20 (Thursday), and 0 (Friday). Figure 9.3a plots the spot and futures prices from Examples 9.2 and 9.3, and Figure 9.3b shows the behavior of the basis over time.

9.6

Trading Spreads

Futures traders trade naked futures or establish various hedge positions. They also trade spreads whereby they try to exploit relative movements between two futures prices that usually track one another for economic, historical, and other reasons. One can set up an intracommodity spread in the futures market by simultaneously buying and selling futures contracts on the same commodity but with different maturity months or an intercommodity spread by simultaneously buying and selling futures contracts on different commodities with the same or different delivery months. When delivery dates differ, it is called a time spread or a calendar spread.

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Why trade spreads? If you are placing directional bets with futures, you need to have expectations about future spot prices. Alternatively, market forces sometimes create abnormal differences between two futures prices that should closely track one another. People speculate with spreads in such situations, hoping to profit when the price difference corrects itself. Notice that when trading spreads, there is no risk from the direction in which the futures prices move. Spread trading can be tricky because spreads may vary with the seasons. For example, although since 1970 wheat has on average cost a dollar more per bushel than corn, an article titled “Corn, Wheat Swap Roles as Prices Surge” (Wall Street Journal, August 9, 2011) noted the opposite: corn was trading higher than wheat, a gap that grew to 66.75 cents in July 2011. This reversal was caused by low supply as well as rising demand for corn from China, while wheat prices fell as many countries had bumper crops. If futures prices move together with spot prices, this is an ideal setting for an intercommodity spread trade that sells corn and buys wheat futures. This strategy was adopted by many traders at the time hoping for “wheat gaining on corn, since the main wheat harvest is over and more corn supplies are poised to hit the market.” But many traders also avoided this strategy in 2011 because, as observed an agricultural consultant, “the seasonal reliability is not there.” Still, spreads are usually safer than an outright long or short position. For this reason, brokers tend to require less margin for spread positions than for naked positions. When closing out a futures or an option position, they regularly ask, Is this part of a spread position? The next two examples illustrate hypothetical spread trading when the futures prices are narrower (see Example 9.7) or wider (see Example 9.8) than expected.

FIGURE 9.3A: Spot and Futures Prices Dollars 1,010 1,005 Spot Prices

1,000

Futures Settlement Prices

995 990 985 980 975

Time 1

2

3

4

5

TRADING SPREADS

FIGURE 9.3B: Behavior of Basis Dollars 0

Time 1

2

3

4

5

–0.1 –0.2 –0.3 –0.4 –0.5 –0.6 –0.7

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EXAMPLE 9.7: An Intracommodity Spread ■

Suppose you notice that the April gold futures price is $1,008, whereas that for December is $1,014 (all prices are per ounce). In traders’ jargon, the spread is $6 to the December side. Sensing a mispricing, you consult your extensive collection of past price data and find that this intracommodity spread has historically hovered around $8.

■

Set up a spread trading strategy: buy the relatively underpriced December futures contract and simultaneously sell the relatively overpriced April futures contract (see Figure 9.4). Traders say that you are buying the spread for $6. No money changes hands because you are trading futures, but you have to keep margin deposits with your broker.

■

Suppose that next month, the April futures price stays the same, but the December futures price rises to $1,016, as predicted. Close out your positions by selling the spread for $8: sell December and buy the April gold futures contract. You make no profit on the April contract but $2 on the December. Stated differently, you make $2 by buying the spread for $6 and selling it for $8.

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Example 9.8 considers an intercommodity spread.

EXAMPLE 9.8: An Intercommodity Spread ■

Suppose that the platinum futures price is $1,080 per ounce and the gold futures price for a contract expiring in the same month is $1,000 per ounce.

■

Suppose you become a technical analyst, put on green goggles, and gaze intently at price charts on the computer screen. You note that the charts reveal that the intercommodity spread should narrow to $50. You can speculate on this insight by selling the spread for $80: sell the relatively overpriced platinum futures and buy the relatively underpriced gold futures contract.

■

Gold futures prices can rise, stay level, or fall, and so can platinum futures prices. Suppose that a month from now, the spread magically narrows to $50. You earn a profit of $30

FIGURE 9.4: Trading Intra-Commodity Spread Today

Next Month Dec futures price $1,016

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December futures price $1,014 Spread $6

Spread $8

April futures price $1,008

April futures price $1,008

Buy Dec and sell April (“buy spread for $6”)

Sell Dec and buy April (“sell spread for $8”)

When trading spreads, remember the wry warning (attributed to renowned economist John Maynard Keynes) that “the market can remain irrational longer than you can remain solvent.”

9.7

Summary

1. One can initiate a futures position by calling a broker and placing an opening order. There are many order placement strategies, the simplest being a market order that is immediately executed. One can move out of a futures position by (1) a closing transaction, (2) physical delivery, (3) cash settlement, or (4) an exchange for physicals. 2. To open a futures position, one must keep an initial margin to guarantee contract performance. When a futures price goes up, a long position gains and a short

QUESTIONS AND PROBLEMS

position loses profits. Through a process called daily settlement, the long’s account is credited with variation margin (which is the change in the settlement price from the previous trading day) and the short’s account is debited the same. The opposite happens when a futures price goes down—long loses and short makes a profit. After daily settlement adjustments are made, the brokerage account is said to be marked-to-market. This substantially lowers credit risk and makes futures safer to use. 3. Futures prices have a number of useful properties. For example, the futures price must equal the spot price at the contract’s maturity. As the contract maturity approaches, the basis (= Spot – Futures price) converges to zero. In an ideal setting (when market frictions are zero), forward and futures prices are equal when interest rates are constant, but unequal when interest rates are random.

9.8

Cases

Investment Linked to Commodity Futures (Harvard Business School Case 2930-

17-PDF-ENG). The case examines an investment linked to an index of commodity futures prices and explores how the index is constructed, how commodity futures behave, and what the portfolio impacts of such an investment might be. Futures on the Mexican Peso (Harvard Business School Case 296004-PDF-ENG).

The case considers the issues that the Chicago Mercantile Exchange faces regarding how to design, and whether and when to introduce, a futures contract on the Mexican peso. Copyright © 2018. World Scientific Publishing Company Pte. Limited. All rights reserved.

Alcoma: The Strategic Use of Frozen Concentrated Orange Juice Futures

(Harvard Business School Case 595029-PDF-ENG). The case explores price risk management in the orange juice industry when an increase in orange tree production led to a surplus production of orange juice.

9.9

Questions and Problems

9.1. What is a closing transaction on a futures contract? Do you need to take delivery

to terminate a futures contract position? 9.2. What does marking-to-market mean? 9.3. What risk do margin accounts on futures contracts and marking-to-market

minimize? Explain your answer. 9.4. What is the difference between initial margin and variation margin of a futures

contract? Provide a simple example to explain your answer. 9.5. Explain what a margin call is and when it occurs on a futures contract. 9.6. Ignoring credit risk, if interest rates are constant and equal to zero, do forward

prices equal futures prices? Why?

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9.7. What is the basis of a futures contract? Give a simple example to explain your

answer. What does convergence of the basis mean with respect to a futures contract? 9.8. Ignoring credit risk, when interest rates are random, must forward prices and

futures prices be equal? Explain why or why not. 9.9. Suppose you go short one contract of gold at today’s closing futures price

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of $1,300of $1,300 per ounce. Suppose that your brokerage firm requires an initial margin of 5 percent of the position size ($130,000 in this example) and sets the maintenance margin at 80 percent of the initial margin. Contract size is one hundred ounces. Keep the margins constant throughout this example. Track the value of your margin account if the closing futures prices are as follows. Clearly identify any margin call received and the amount of variation margin that you have to produce. Day

Closing Futures Price (dollars/ounce)

0 (today)

$1,300

1

1,303

2

1,297

3

1,290

4

1,297

The next two questions are based on the following data: The euro FX (currency) futures contract that trades in the Merc has the following features:

Euro FX Contract Highlights Ticker symbol

EC

Trading unit (underlying)

Euro 125,000

Quotation

US$ per euro

Minimum price fluctuation (tick)

$.0001/euro = $12.50/contract

Contract months

March, June, September, December

Assume initial margin is $2,511 and maintenance margin is $1,860 per contract (the exchange periodically revises these numbers). 9.10. Track margin account payments to a trader holding long position in two

contracts when the euro takes the following values (in terms of US dollars): 0.8450, 0.8485, 0.8555, 0.8510, 0.8480, 0.8423, 0.8370, 0.8300, 0.8355.

QUESTIONS AND PROBLEMS

9.11. Track the margin account payments to a trader holding short position in one

contract when the euro takes the following values (in terms of US dollars): 0.8450, 0.8485, 0.8555, 0.8510, 0.8480, 0.8423, 0.8370, 0.8300, 0.8355. 9.12. Suppose that the platinum futures price is $1,580 per ounce and the gold futures

price for a contract expiring in the same month as platinum is $1,500 per ounce (see Example 9.8). Hoping that the spread will narrow to $50 in a month’s time, set up a spread trading strategy, discuss all possible outcomes for the futures prices after a month, and illustrate these outcomes in a diagram. 9.13. Explain the difference between cash-settled and physical-settled futures

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contracts.

191

10 Futures Regulations 10.1 Introduction 10.2 Which Markets Have Futures? Price Discovery

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Hedging Speculation EXTENSION 10.1 When Sellers Have Market Power

10.3 Regulation of US Futures Markets Major Regulatory Acts The CFTC, the NFA, and the Regulatory Role of Exchanges

Enforcement Actions by the CFTC and the NFA

10.4 Manipulation in Futures Markets The Hunts Silver Case Salomon Brothers and US Treasury Securities Banging the Close

10.5 Managing Commodity Markets 10.6 Summary 10.7 Cases 10.8 Questions and Problems

WHICH MARKETS HAVE FUTURES?

10.1

Introduction

Price discovery is an important function of futures markets. Even so, US citizens became upset in July 2003 when the Defense Advanced Research Projects Agency (DARPA), a research wing of the US Department of Defense, announced plans for a “Policy Analysis Market,” where anonymous individuals could trade futures whose payoffs would depend on a terrorist indicator. The proposal that the US government set up an online betting parlor for terrorist activities enraged both legislators and the public. The project was killed soon after it was unveiled. Despite its ethical problems, it was hailed as a good idea by some economists, who noted that futures prices have done an excellent job of predicting conventional as well as nonconventional events. DARPA argued that futures markets usually fare better than experts in predicting events like elections, and this project’s mission was to tap the collective wisdom in market prices—price discovery—to understand the probabilities of terrorist activities. But this argument from the Department of Defense didn’t quite fly. This chapter discusses the price discovery role of futures markets, regulations, and market manipulation. First, we explore the market conditions needed for futures to trade. Next, we look at futures market regulations. Finally, we discuss futures trading abuses and market manipulation: colorful stories of powerful players who sought quick gains by cornering the market and squeezing the shorts until regulators foiled their ploys.

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10.2

Which Markets Have Futures?

When is a futures market for a particular commodity likely to succeed? Three conditions—competitive markets, standardization, and volatile prices—seem to be among the key ingredients needed. A competitive market is one in which each investor’s trades are too insignificant to influence prices and traders act as price takers (see Varian 2003). Economists view competitive markets as the ideal setting. Markets cease to be competitive when some participants gain market power and manipulate prices to their benefit. When markets can be manipulated, traders are reluctant to trade knowing that they can lose significant wealth to the manipulators. For example, consider the following: ■

When a monopoly (a single seller), an oligopoly (a small band of sellers), or a cartel fixes commodity prices, futures trading may not develop. This is the case for diamonds but not for oil. Extension 10.1 tells a tale of the spectacularly successful diamond cartel.

■

When a sole buyer or a small number of buyers drive the market, futures markets for the commodity are unlikely. There are no futures markets for missiles and weapons where the government is the sole buyer.

Standardization refers to the ability of a commodity to be sold in standardized units. For example, gold and silver are easy to standardize in terms of quantity

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and quality. As such, one can easily write contracts on future purchases or sales of these commodities in large quantities. In contrast, automobiles are not so easily standardized. There are hundreds of different car models available, distinguished by manufacturer and other features like manual or automatic, electric, gas, or diesel, and so on. Consequently, it is no surprise that automobile futures do not exist. Volatile prices are commodity prices that change randomly across time in an erratic (up and down) fashion. Volatile prices create risk for producers in determining both their input costs and output sales. Volatile prices therefore generate a demand to hedge these price risks and also a need to forecast future prices. Futures markets exist to fulfill these two needs. Let us explore both these roles filled by futures markets, price discovery, and hedging in more detail.

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Price Discovery Price discovery is related to the notion of market efficiency (see Chapter 6). In efficient markets, prices accurately reflect available information. Such information includes past and current prices (weak-form efficiency), publicly available information (semi-strong-form efficiency), and private information (strong-form efficiency). This concept is not only true for spot markets but for futures markets as well. Because futures market traders include producers who have private knowledge of supply and demand conditions, futures markets are widely believed to be (semistrong-form) efficient, thereby generating prices that reflect more information and provide better price forecasts—price discovery—than any ordinary trader’s information could. This is the motivation underlying the US Department of Defense’s July 2003 proposal to set up a futures market in terrorism indicators. Although well intentioned, the plan was shelved following heavy criticism concerning its ethical considerations. Price discovery helps futures buyers and sellers of commodities. For example, firms that process and distribute corn can fix the buying and/or selling price by trading futures, or the retail seller, which may be your neighborhood supermarket chain, can do the same.

Hedging Hedging in the New Oxford Dictionary of English is defined as to “protect (one’s investment or an investor) against loss by making balancing or compensating contracts or transactions.” If you hedge with futures (for which you pay nothing to start), then you get protection from price swings. You benefit when the spot adversely moves but surrender gains when the spot moves in your favor. Hedging with futures is beneficial to firms for minimizing input and output price risks, thereby smoothing costs and profits. This is true for both producers (e.g., farmers, manufacturers) and suppliers (e.g., gold and copper mines). Hedging is the motivation for the existence of oil futures markets, despite that the oil producers form a pricing cartel known as the Organization of the Petroleum Exporting Countries (OPEC).

WHICH MARKETS HAVE FUTURES?

195

Speculation In contrast to hedging, the term speculation carries a negative connotation. The humorist Mark Twain said, “There are two times in a man’s life when he should not speculate: when he can’t afford it, and when he can.” But speculation is really the flip side of hedging. Hedgers are buying insurance against adverse price movements. Speculators are selling the insurance: gambling that the adverse price movements will not occur. For futures markets to work, both hedgers and speculators are usually needed, and both play a crucial role. The reason is that it is unusual for hedgers on both sides of the market (suppliers and users) to have exactly offsetting demands. As such, hedging demands for a commodity are often one-sided. In such circumstances, speculators provide the liquidity needed for a successful futures market to exist. In addition to their role in providing market liquidity—being the insurance sellers—speculators also play a second important role in futures markets: they facilitate the price discovery role because they often trade based on private information, obtained by costly investigation. Speculators trading on the basis of their private information increase market efficiency because as they trade, their information gets reflected in market prices. Without speculators, futures markets would be both less liquid and less efficient, and sometimes, without speculators, futures markets would not exist. An example is the market for precious diamonds, in which the Diamond Cartel removed the speculators’ incentives to participate (see Extension 10.1).

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EXTENSION 10.1: When Sellers Have Market Power

Cartels An oligopoly is a market that is dominated by a handful of sellers, more than one but not too many. For example, Boeing of the United States, Airbus of Europe, and some fringe suppliers like Ilyushin and Tupolev (now divisions of Russia’s United Aircraft Corporation) dominate the production of large passenger airplanes. Members of an oligopoly sometimes join to form a cartel. A cartel is a coalition of producers that monopolizes a commodity’s production and sale. A cartel tries to maximize joint profits by controlling supply—it allocates a share of the common market to each member, sets sales quotas, regulates production, and fixes prices. By eliminating competition, a cartel raises prices for consumers and profits for the producers. In the process, it may keep inefficient companies in business. Cartels are banned in the United Kingdom and the United States. However, they are legal in many parts of the world, including highly industrialized nations like France, Germany, Italy, and Japan. By controlling the supply of a commodity, and therefore prices, cartels can potentially eliminate the benefits of both hedging and price discovery in futures markets. In such circumstances, futures markets will not exist. This is the case with the Diamond Cartel.

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De Beers and the Diamond Cartel

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Cecil Rhodes founded this enterprise during the 1880s by consolidating several diamond mines in Southern Africa. Around the same time, he formed the London Diamond Syndicate to control the world’s diamond trade. This was later transformed by Sir Ernest Oppenheimer into the Central Selling Organization, a group of marketing organizations that controlled much of the world’s diamond trade until the last decade of the twentieth century. A February 2001 Fortune magazine article (“The De Beers Story: A New Cut on an Old Monopoly”) reported that De Beers entered agreements with most major diamond producers (including the former Soviet Union) and led a very effective cartel. In the 1990s, about 45 percent of the global diamond production was mined by De Beers and another 25 percent went through its hands. In 2000, De Beers announced that it would stop manipulating the supply by buying and storing diamonds. Afterward, De Beers focused on selling diamonds from its own mines after branding them with a “Forevermark” to guarantee their integrity. These steps strengthened the demand for diamonds à la De Beers. Despite this announcement, the effectiveness of the diamond cartel did not disappear. A New York Times article dated August 9, 2008, titled “Talking Business: Diamonds Are Forever in Botswana,” reported that “today, De Beers has about 40 percent of the diamond market—but it is far more profitable than under the old regime, when it controlled 80 percent of the market.” The strong control of the supply of diamonds by the De Beers Group destroyed the price discovery role of a diamond futures market as documented because diamond prices do not reflect supply and demand but the activities of De Beers. Although hedging demands certainly exist for firms using diamonds in production, there are no offsetting hedging demands from the suppliers (the cartel). For a diamond futures market to work, speculators need to provide the missing liquidity, but the strong control of the market supply by De Beers removed the desire for speculators to participate in such a market. Consequently, no diamond futures markets exist today.

10.3

Regulation of US Futures Markets

Futures markets are not for the fainthearted. Wild price swings have been common since inception, and there have been numerous attempts to manipulate futures prices. Competitive markets can lead to economic growth as well as efficiency. However, sometimes they fail to function, creating a need for regulation. As is often the case with regulations, many of the rules and regulations of the US securities market have arisen to fix problems evidenced within financial markets during a financial crisis.

Major Regulatory Acts Regulation of the US futures markets began on a cold winter’s day in February 1859, when the governor of Illinois signed a legislative act that granted a corporate charter to the Chicago Board of Trade (CBOT). The charter, among other things, granted the exchange self-regulatory authority over its members, standardized commodity grades,

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REGULATION OF US FUTURES MARKETS

and provided for CBOT-appointed grain inspectors whose decisions were binding on members. This was a critical early step in the evolution of futures contracts.1 Another major step was taken in October 1865, when formal trading rules were introduced at the CBOT, including margin requirements and delivery procedures. Three years later, the exchange adopted a rule to deter manipulation by banning “‘corners’ (defined as ‘making contracts for the purchase of a commodity, and then taking measures to render it impossible for the seller to fill his contract, for the purpose of extorting money from him’).”2 In 1877, the CBOT began publishing futures prices on a regular basis, and in 1883, the first organization was formed to clear CBOT contracts on a voluntary basis. US federal regulations with respect to futures markets began in the 1880s when the first bills “to regulate, ban, or tax futures trading in the U.S.”3 were introduced in Congress. About two hundred such bills were introduced over the next forty years (see Table 10.1 for a list of some major US futures regulations). As agricultural futures dominated the markets in the early days, regulations primarily focused on grain (such as corn, wheat, oats, and rye) trading, and the regulatory controls were placed in the hands of the US Department of Agriculture (USDA). In 1922, the Grain Futures Act was passed. It regulated grain futures trading, banned “off-contract-market futures” trading, created (an agency of the USDA) the Grain Futures Administration for administering the act, and established the Grain Futures Commission (consisting of the secretaries of agriculture and commerce and the attorney general), which had the broad power of suspending or revoking a contract market’s designation. The Commodity Exchange Act of 1936 amended and extended the 1922 act. It replaced references to “grains” with the term “commodities,” and it expanded the approved list to include commodities like butter, cotton, and eggs. The Grain Futures Commission became the Commodity Exchange Commission and was granted the authority to establish federal speculative position limits. In 1947, this commission evolved into the Commodity Exchange Authority (also an agency of the USDA). It continued to administer the Commodity Exchange Act until the mid-1970s. But all this changed with the passing of the Commodity Futures Trading Commission (CFTC) Act of 1974. This created the CFTC as the independent, ultimate “federal regulatory agency for futures trading,” taking responsibility for regulating futures trading in 1975. The CFTC Act also proposed creation of a self-regulatory organization. Subsequently, an industry body called the National Futures Association (NFA) was established in 1982, which works with the CFTC to help regulate the markets. More legislation followed in 1978 and 1982. This whole body of legislation is known as the amended Commodity Exchange Act (CEA).

1

See “History of the CFTC: US Futures Trading and Regulation Before the Creation of the CFTC” from www.cftc.gov/About/HistoryoftheCFTC/history_precftc. 2

ibid.

3

ibid.

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TABLE 10.1: Some Major Milestones in the Federal Regulation of US Futures Markets Year

Development

1880s

The first bills are introduced in Congress to regulate futures markets.

1922

The Grain Futures Act was passed. The Grain Futures Administration was created as a department of the US Department of Agriculture.

1936

The Commodity Exchange Act, an amendment of the 1922 Act, was passed. The Commodity Exchange Administration was created as the first federal regulator of futures markets.

1974

The Commodity Futures Trading Commission Act was passed. This created the Commodity Futures Trading Commission as an independent “federal regulatory agency for futures trading.”

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1978, 1982

The Futures Trading Acts were passed.

1982

The National Futures Association, a self-regulatory agency for the futures industry, was established.

2000

The Commodity Futures Modernization Act of 2000 was passed.

2010

The Dodd–Frank Wall Street Reform and Consumer Protection Act was passed. This significantly increased regulatory oversight of the over-the-counter derivatives market.

A landmark legislation that substantially altered the CEA, the Commodity Futures Modernization Act of 2000 (CFMA), was passed by Congress and signed into a law in December 2000. The major changes included the following: 1. Lifting of a long-standing ban by the CFMA, which made it possible to trade security futures products, which are futures contracts based on single securities and narrowly based stock indexes. 2. Easing up on regulations for swaps, other over-the-counter (OTC) derivatives, and trading facilities for these products. 3. Streamlining of regulation and the creation of three types of markets: (1) designated contract markets (DCMs, which are boards of trade [or exchanges] that operate under the regulatory oversight of the CFTC), (2) derivative transaction execution facilities, which have fewer regulations but more restrictions on who and what trades, and (3) exempt boards of trade with restrictions similar to (2) but exempt from CFTC regulation, except for antifraud and antimanipulation provisions. The CFMA regulations reflected the deregulatory mood prevalent at the time. The loosely regulated OTC derivative markets seemingly worked well until the financial

REGULATION OF US FUTURES MARKETS

crisis of 2007–9, where it was documented that trading in OTC derivatives helped cause the crisis. In response to this lack of regulation, in 2010, Congress passed the Dodd–Frank Wall Street Reform and Consumer Protection Act. This massive 849-page act4 was the largest effort to regulate US financial markets since the 1930s Great Depression. It introduced numerous significant changes to bank and financial market regulation. Some key features of the Dodd–Frank Act related to derivatives are as follows:5 ■

Increased regulation. Provides the SEC and CFTC authority to regulate OTC derivatives so that irresponsible practices and excessive risk taking no longer escape regulatory oversight.

■

Central clearing and exchange trading. To require central clearing and exchange trading for more derivatives, this act replaced derivatives transaction execution facilities with a new type of entity, the swap execution facility.

■

Market transparency. This requires data collection by clearinghouses and swap repositories to improve market transparency and provide regulators the necessary information for monitoring and responding to systemic risks.

■

Financial safeguards. These add safeguards by ensuring that dealers and major swap participants have adequate financial resources to guarantee the execution of derivatives contracts.

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The CFTC, the NFA, and the Regulatory Role of Exchanges In establishing the CFTC, Congress recognized that futures markets are important to economic growth. The CFTC carries out a variety of activities (see https://www.cftc.gov/About/CFTCOrganization/index.htm): ■

The CFTC must approve new contracts and changes to existing contracts. Each new futures contract is evaluated with respect to three issues: (1) justification of individual terms and conditions, which thoroughly examines the details of the contract to ensure that there will be sufficient supply of the underlying commodity to prevent market manipulation; (2) an economic purpose test, which requires exchanges to show that the proposed contract will be useful for price discovery and hedging; and (3) other public interest requirements.

■

The CFTC develops rules and regulations that govern the NFA and all futures exchanges. These rules and regulations define the requirements for registration, disclosure, minimum financial standards, daily settlement, separation of customer funds, supervision and internal controls, and other activities.

4 5

See www.gpo.gov/fdsys/pkg/PLAW-111publ203/pdf/PLAW-111publ203.pdf.

See http://banking.senate.gov/public/_files/070110_Dodd_Frank_Wall_Street_Reform_comprehensive_summary_Final.pdf.

199

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■

The CFTC requires companies and individuals who handle customer funds or give trading advice to register with them. Actually, the NFA handles the registration process on the CFTC’s behalf.

■

The CFTC ensures compliance of its rules and regulations as well as the Commodity Exchange and Dodd–Frank acts. It conducts trade practice surveillance and audits selected registrants. It investigates and prosecutes alleged violations of CFTC regulations. Whenever necessary, the CFTC files court cases.

■

The CFTC detects and prevents manipulation, congestion, and price distortions. It conducts daily market surveillance. It can intervene and take corrective action if it believes manipulation is present.

■

The CFTC oversees training of brokers and their representatives and other industry professionals. It can force brokers and representatives to take competency tests. The CFTC also makes sure that all registrants complete ethics training.

■

The CFTC does research on futures markets and provides technical assistance. It makes economic analyses for enforcement investigations and gives expert help and technical aid with case development and trials to US attorney’s offices, other federal and state regulators, and international authorities.

■

The CFTC helps coordinate global regulatory efforts. It develops policies and regulations governing foreign and cross-border transactions.

■

The CFTC handles customer complaints against registrants. It hears and decides enforcement cases. The CFTC offers a reparations procedure for persons who have reason to believe that they have suffered a loss due to a violation of the Commodity Exchange Act or CFTC regulations in their dealings with a CFTC registrant.

The SEC regulates securities and the CFTC regulates futures, but who regulates options? Following an intense turf battle between these two regulators, the agreement is that the SEC regulates options on stocks, while the CFTC regulates those on futures. The fight relapses whenever new derivatives are introduced that carry joint characteristics. The issue of overlapping jurisdiction remains unresolved at this time. On the next rung of the regulatory ladder are the commodity exchanges and the National Futures Association. These self-regulatory organizations (SROs) must, among other things outlined in the CFTC’s regulations, enforce minimum financial and reporting requirements for their members. As a registered futures association under the Commodity Exchange Act, the NFA is an SRO composed of futures commission merchants, commodity pool operators, commodity trading advisors, introducing brokers, leverage transaction merchants, commodity exchanges, commercial firms, and others who work in the futures industry.6 Banks and exchanges may join the NFA, but they are not required to do so. The NFA is involved in a range of activities that include the following:

6

www.nfa.futures.org.

REGULATION OF US FUTURES MARKETS

■

On behalf of the CFTC, the NFA registers all categories of persons and firms dealing with futures customers.

■

The NFA screens and tests registration applicants and determines their qualifications and proficiency.

■

The NFA requires futures commission merchants (FCMs) and introducing brokers to keep sufficient capital and maintain good trading records.

■

The NFA tracks financial conditions, retail sales practices, and the business conduct of futures professionals, and it ensures compliance with the requirements.

■

The NFA audits, examines, and conducts financial surveillance to enforce compliance by members.

■

The NFA maintains an arbitration program that helps to resolve trade disputes.

Commodity exchanges complement federal regulations with their own rules. These rules cover many aspects of futures trading such as the clearance of trades, trade orders and records, position limits, price limits, disciplinary actions, floor trading practices, and standards of business conduct. Clearinghouses also take part in regulatory activities.

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Enforcement Actions by the CFTC and the NFA The NFA has comprehensive compliance rules “covering a wide variety of areas such as advertising, telephone solicitations, risk disclosure, discretionary trading, disclosure of fees, minimum capital requirements, reporting and proficiency testing.” The NFA takes disciplinary action against violators, ranging from issuing warning letters to filing formal complaints that can result in penalties that include “expulsion, suspension for a fixed period, prohibition from future association with any NFA Member, censure, reprimand and a fine of up to $250,000 per violation.” The CFTC’s Division of Enforcement investigates and prosecutes alleged violations of the Commodity Exchange Act and Commission regulations. The CFTC takes enforcement action against individuals and firms registered with the commission, those who are engaged in commodity futures and option trading on designated domestic exchanges, and those who improperly market futures and options contracts. The CFTC also vigorously investigates and prosecutes Ponzi schemes. Named after the notorious swindler Charles Ponzi, a Ponzi scheme is a scam that pays early investors returns from the cash coming in from subsequent investors. A common technique is to start a commodity pool and lure unwary, unsophisticated investors by promising them a generous return. Eventually, the whole scheme unravels and leaves behind a trail of cheated customers. Ponzi schemes have attracted special attention since 2008, when it was revealed that prominent stockbroker and financial adviser Bernard Madoff ran a giant Ponzi scheme that bilked investors for tens of billions of dollars.

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10.4

Manipulation in Futures Markets

In 1869, notorious financiers Jay Gould, James Fisk, and their co-conspirators tried to corner the gold market. As told in “Gold at 160. Gold at 130” (Harper’s Weekly, October 16, 1869), the conspirators bought up huge amounts of gold and attempted a market corner. The result was a two-week period of turmoil in the gold market that brought America’s foreign trade, which used gold as the medium of exchange, to a virtual standstill. To defuse a looming economic panic and restore normal conditions in the gold market, President General Ulysses Grant ordered the sale of gold to buy bonds. On September 24, 1869, minutes after the Treasury secretary’s telegram instructing a sale of $4 million worth of gold reached the “Gold Room,” the price of gold, which had earlier ranged between $160 and $162, fell to $133 per ounce. The day went down in history as Black Friday, and the speculators got a severe jolt. This was just one incident. As we noted in Chapter 6, market corners and short squeezes have a long history in financial markets. Market structure, the creation of shorts as futures get transacted, and supply variability for many of the underlying commodities make futures markets especially prone to price manipulation and trading abuses. We now discuss some well-known cases of manipulation in these markets.

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The Hunts Silver Case A Peruvian government firm Minpeco SA sued the Hunt brothers and “coconspirators (including Arab sheiks)” for $150 million, charging that they conspired to corner the silver market. A jury found them guilty, and the Hunt brothers were given huge fines. A class-action suit was also filed on behalf of seventeen thousand investors. Eventually, in the late 1980s, two of the Hunt brothers filed for bankruptcy. The details were as follows (see Duffie [1989], pp. 335–9 and Edwards and Ma [1992], p. 198–9): ■

Oil magnate H. L. Hunt, one of world’s richest persons, had two wives. One of his wife’s sons (referred to here as the Hunt brothers or the Hunts) had a mindboggling net worth of over $5 billion in 1980 but ended bankrupt after the Silver Crisis.

■

Several times during the 1970s, the Hunts took large long positions in both silver spot and futures markets and rolled their hedge forward (i.e., closed out a position in an expiring futures and then took an identical position in a distant-month futures contract). They also invested in a number of silver producers.

■

This pattern was repeated in 1979–80 on a grander scale. The Hunt brothers (1) held large long positions in silver futures (at times reaching nearly 250 million ounces—an amount equivalent to fifty thousand COMEX contracts [that traded in the Commodity Exchange] and much more than US domestic silver consumption), (2) held dominant long positions in many near-month contracts, (3) demanded delivery when contracts matured, and (4) made supply scarce by holding huge quantities of silver bullions and coins. Some twenty futures commission merchants helped the Hunt brothers, and the firms Bache Halsey Stuart Shields and Merrill Lynch handled over 80 percent of the Hunt brothers’ spot and futures trading.

MANIPULATION IN FUTURES MARKETS

■

Silver prices shot up. Historically, silver prices rarely crossed $10 per ounce, but the spot and futures price of silver rose from about $9 per ounce in July 1979 to $35 per ounce at year’s end, peaking at over $50 per ounce in January 1980. Apparently, the Hunts had cornered the market.

■

The exchanges consulted the CFTC and took steps to break the squeeze. Margin requirements were increased, stricter position limits were imposed, and the Hunts were forced to implement liquidation-only trading. They could not increase their silver position unless it was for hedging purposes. They also had to exit the market as contracts expired. By the end of March, silver prices dropped to $11 per ounce. It was this decline in silver prices that destroyed the Hunts’ wealth.

Salomon Brothers and US Treasury Securities

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The US Treasury securities market is one of the world’s largest and most liquid securities markets. Legal scholars like Easterbrook (1986) and Fischel and Ross (1991) have opined that it is nearly impossible to corner the supply of Treasuries. Unfortunately, the impossible occurred as Salomon Brothers Inc., a major player in the Treasury securities market, admitted deliberate and repeated violations of Treasury auction rules beginning in 1990. Here is how the Salomon short squeeze in the May 1991 auction of the two-year US Treasury note worked: ■

There are three related markets for buying Treasuries: (1) the sealed bid Treasury auction, (2) the preauction when-issued market, and (3) the postauction resale market (see Chapter 2).

■

Salomon took considerable pride in its trading prowess and boasted itself to be the greatest bond trading firm in the world. To ensure broad participation (read, stop Salomon’s dominance!) in its auction of Treasury securities, the Treasury enacted the 35 percent rule for each auction: a single bidder cannot bid more than 35 percent of the offering at any particular yield or be awarded more than 35 percent of the offering.

■

Many Wall Street firms sell Treasuries to their clients in the when-issued market, enter the auction with a net short position, and buy securities in the auction to fulfill their obligations. This is a profitable business because the price usually goes down slightly from the when-issued price owing to greater supply in the auction— but not in this auction. Salomon submitted aggressive bids at a yield of 6.81 percent when the notes were trading on a when-issued basis at approximately 6.83 percent directly prior to the auction (which means Salomon paid more for the auctioned securities).

■

Salomon evaded the 35 percent rule by submitting multibillion dollar bids on behalf of clients and then transferred those securities to its own account at cost. Salomon ended up controlling 94 percent of the competitively auctioned securities. Having cornered the market, Salomon squeezed the shorts from the when-issued market by charging them a premium price for these notes when they came to cover their short positions in the post auction resale market.

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■

Academics have tried to measure how much Salomon stood to gain from this squeeze. Studies estimate that this ranged between $5 million and $30 million (for details, see Jegadeesh 1993; Jordan and Jordan 1996).

■

The penalties to Salomon were enormous. Its reputation was badly bruised. Soon after the cornering allegations surfaced and federal agencies started investigating, regulators barred Salomon from making a market in various federal and state government securities, cutting off a major source of the company’s income. Salomon stock, which was trading in the $50 to $60 range, soon plunged to $30. Billionaire Warren Buffett, a major shareholder of Salomon, flew in from Nebraska and took control. Several top executives lost their jobs, and many talented personnel left. Salomon admitted no wrongdoing other than violating Treasury auction rules and paid a staggering $290 million in fines. The once-feared Salomon now belongs to the past. It went through mergers and acquisitions and is now part of Salomon Smith Barney, which is a subsidiary of Citigroup. Even the name Salomon has been phased out.

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Banging the Close Technology has opened up new ways of manipulating prices. On July 24, 2008, the CFTC filed a complaint against Netherlands-based Optiver Holding, two of its subsidiaries, and three Optiver employees in a US district court, charging that the defendants had manipulated or attempted to manipulate NYMEX crude oil, gasoline, and heating oil futures contracts on nineteen separate instances in March 2007 and succeeded in creating artificial prices in at least five of those cases.7 The defendants traded three NYMEX energy futures contracts: the Light Sweet Crude Oil futures contract (Crude Oil, also referred to as West Texas Intermediate [WTI]), the New York Harbor Heating Oil futures contract (Heating Oil), and the New York Harbor Reformulated Gasoline Blendstock futures contract (New York Harbor Gasoline). The CFTC alleged that the defendants employed a practice popularly known as banging (or marking) the close, which involves manipulating the prices by trading a large position leading up to the close followed by offsetting the position before the end of trading. The scheme had the following ingredients: ■

It involved Trading at Settlement (TAS) contracts for crude oil, heating oil, and gasoline contracts. These are special futures contracts in which the counterparties decide at the time of trading that the contract price will be the day’s settlement price plus or minus an agreed differential. If you trade a TAS contract, you can offset it by trading a futures contract on the other side of the market.

■

Computing settlement prices is tricky in many futures markets because of potential manipulation. The settlement prices for each of these three energy futures contracts

7

Our discussion is based on “Complaint: Optiver US, LLC, et al.,” “CFTC Charges Optiver Holding BV, Two Subsidiaries, and High-Ranking Employees with Manipulation of NYMEX Crude Oil, Heating Oil, and Gasoline Futures Contracts” and “Case Background Information re CFTC v. Optiver US, et al.” (www.cftc.gov/PressRoom/PressReleases/ pr5521-08).

MANAGING COMMODITY MARKETS

are determined using the volume-weighted average prices (VWAP) of futures trades occurring during the closing period (the close) for the contracts, which lasts from 2:28 to 2:30 PM. The defendants planned to control the VWAP and wrote about “bullying the market” in their recorded conversations and e-mails. ■

The traders concocted a three-step system to make manipulative profits: - First, they would accumulate substantial positions in TAS contracts by trading at different times of the trading day. - Second, they would enter closing trades for approximately 20 to 30 percent of their contracts during the pre-close period, which are the dying minutes before the market enters the close. - Third, Optiver would sell 70 to 80 percent of its holdings during the close, which lasts from 2:28:00 to 2:29:59. In conjunction, these three steps manipulated prices so that Optiver could obtain profits from its trades.

■

The CFTC alleged that the defendants had forced the futures prices lower on three instances and higher in two cases and made approximately $1 million in the process.

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10.5

Managing Commodity Markets

The economic definition of a speculator is someone who is trading and not hedging. For example, a trader who is short gold futures but does not hold gold in inventory is speculating. In contrast, if this trader holds gold in inventory for production purposes, then she is a hedger. Futures regulations regard speculators as “evil” and impose size limits on their positions—to preclude the accumulation of market power for the purposes of manipulative trading strategies. In contrast, they view hedgers as “saintly” and impose no such size limit restrictions. Although the economic definition of a speculator and hedger is unambiguous, in practice, it is not always so easy to tell speculators from hedgers. Prior to the 1990s, the CFTC and exchanges classified futures traders like farmers, manufacturers, and commodities dealers (with legitimate hedging needs) as commercial traders, and the CFTC allowed them to trade virtually any number of exchange-traded futures contracts desired. These were the hedgers. By contrast, they imposed speculative position limits on all other traders, who were viewed as noncommercial traders. These were the speculators. Obviously, noncommercial traders could be hedgers using the economic definition, but they were not considered as such. However, beginning in the 1990s, two new kinds of traders, commodity index traders and swap dealers, entered the markets and changed these traditional classifications. Commodity index traders include institutional players like endowment funds and pension funds. They diversify their portfolios by adding commodity exposures through passive long-term investment in commodity indexes. Swap dealers play a more complex role—on one hand, they compete with the futures markets by offering customized derivatives to their clients, but on the other hand, they

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also trade in the futures markets to hedge risks in their swap books. Typically affiliated with a bank or other large financial institution, swap dealers operate as market makers by being ready to act as the counterparty to both commercial and noncommercial traders. A change in the regulatory framework took place in 1991, when the CFTC granted a Goldman Sachs subsidiary J. Aron the same exemption from speculative position limits as those allowed to commercial traders (see “A Few Speculators Dominate Vast Market for Oil Trading,” Washington Post, August 21, 2008). J. Aron was into commodity merchandising and traded swaps as a part of its business. It planned to sell to a large pension fund a commodity swap based on an index that included wheat, corn, and soybeans, all of which fell under federal speculative position limits. To hedge this short commodities exposure, it planned to buy exchangetraded futures contracts on those commodities. The CFTC classified this as a bona fide hedge because the swap dealer had demonstrated that the positions were “economically appropriate to the reduction of risk exposure attendant to the conduct and management of a commercial enterprise.” Once the door was opened, more exemptions followed.

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10.6

Summary

1. Futures contract trading in the United States is a highly regulated activity. It is governed by the Commodity Exchange Act. The Commodity Futures Modernization Act of 2000 made substantial changes to the existing legislation. It made it possible to trade security futures products, it eased up on regulations of OTC derivatives, it allowed new types of futures trading facilities and products, and it facilitated clearing of OTC derivative trades. In 2010, the Dodd-Frank Wall Street Reform and Consumer Protection Act was passed to increase OTC derivative market regulation in response to the financial crisis of 2007–2009. 2. Market regulation in the United States is implemented at the highest level by the federal regulatory agency, the Commodity Futures Trading Commission, and then by a self-regulatory organization, the National Futures Association, the exchanges themselves, and finally, at the grassroots level, by brokers handling customer accounts. 3. The Commodity Futures Trading Commission monitors the markets, investigates and prosecutes alleged violations of the Commodity Exchange Act and commission regulations, and punishes violators with fines and (temporary or permanent) banishment from the futures industry.

10.7

Cases

Amaranth Advisors: Burning Six Billion in Thirty Days (Richard Ivey School of

Business Foundation Case 908N03-PDF-ENG, Harvard Business Publishing). The case provides students with (1) a deeper understanding of commodity futures markets in general and of natural gas markets in particular, (2) an introduction to

QUESTIONS AND PROBLEMS

hedge funds and an insight into the largest hedge fund collapse in history, and (3) an introduction to such concepts as liquidity risk, value-at-risk, spread trades, and the use of derivatives. Mylan Lab’s Proposed Merger with King Pharmaceutical (Harvard Business

School Case 209097-PDF-ENG). The case considers how hedge funds and other investors may use derivatives to separate votes from shares and the legal, moral, and economic implications of this ability. Rogue Trader at Daiwa Bank (A): Management Responsibility under Different Jurisprudential Systems, Practices, and Cultures (University of Hong Kong

Case HKU442-PDF-ENG, Harvard Business Publishing). The case examines the importance of complying with regulations in the context of a major foreign bank operating in the United States.

10.8

Questions and Problems

10.1. Give two reasons why futures markets may not be started on a commodity.

Give an example of such a commodity to support your answer. 10.2. What is price discovery with respect to the trading of a futures contract? Give

an example of its use. 10.3. Explain why hedging and speculation are like “two sides of the same coin.” 10.4. What is the CFTC’s mission? 10.5. What is the National Futures Association and how does it differ from the Copyright © 2018. World Scientific Publishing Company Pte. Limited. All rights reserved.

CFTC? 10.6. If regulation is bad for business, why does the National Futures Association

put a significant amount of regulation on its members? 10.7. Suppose that you are an economist working for the CFTC and an exchange

has proposed to introduce futures trading on individual stocks. Would you accept this proposal? Give reasons for your answer. 10.8. Suppose that it is five minutes to the close of trading for the day and you

are a trader on the exchange floor. Your client gives you a sell order for one thousand contracts. You are holding one hundred contracts long. How can you take advantage of this situation? 10.9. Summarize what happened in the Hunts silver case. 10.10. Summarize what happened in the Salomon Brothers manipulation. 10.11. Why is the possibility of manipulation bad for the trading of futures contracts?

Give an example of what can go wrong. 10.12. Is selling an insurance contract speculating? Is buying an insurance contract

hedging? Are insurance contracts a benefit to society? Explain why or why not.

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10.13. Explain why speculation in futures trading is analogous to selling insurance

contracts. Does this imply that speculation with futures is beneficial or harmful to the economy? Explain your answer. 10.14. Do futures contracts need to be regulated? If yes, what can go wrong without

regulation? Give an example to explain your answer. 10.15. Can you identify common features in the manipulation stories involving

forward or futures markets? How can “authorities” intervene and break squeezes in a typical situation?

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10.16. What is banging the close?

11 The Cost-of-Carry Model 11.1 Introduction 11.2 A Cost-of-Carry Example 11.3 The Assumptions Copyright © 2018. World Scientific Publishing Company Pte. Limited. All rights reserved.

11.4 The Cost-of-Carry Model The Model Setup Using the Law of One Price Using Nothing Comes from Nothing Different Methods for Computing Interest

The Arbitrage Table Approach

11.5 Valuing a Forward Contract at Intermediate Dates 11.6 Linking Forward Prices of Different Maturities 11.7 Summary 11.8 Cases 11.9 Questions and Problems

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11.1

Introduction

To trade a forward contract on a spot commodity and not get ripped off, one needs to know the fair forward price. But how can one determine this price? We claim that if you understand the concepts of present and future value introduced in chapter 2, then you should already be able to guess the answer. Give up? The answer is that the forward price should be the future value of the spot commodity today! Why? If all’s right with the world, then this should be today’s price for buying the spot commodity in the future. This chapter shows that under a reasonable set of assumptions, including the assumption of no arbitrage studied in chapter 6, this guess is correct. The technique we use to prove this guess is called the cost-of-carry model, and it is the simplest and first application of the arbitrage-free pricing methodology studied in chapter 6. Later chapters will use the same no-arbitrage approach to price more complex derivatives such as options. The cost-of-carry model is derived via a cash-and-carry argument whereby the underlying commodity is purchased with borrowed cash and held until the forward contract’s maturity date, thereby it is carried into the future. When held, various costs-of-carry (or carrying charges) such as interest on the borrowed cash and storage costs for the commodity are paid. At the forward contract’s maturity, the borrowing is repaid, yielding unfettered ownership of the commodity. The cost of purchasing the commodity in the future by the cash-and-carry strategy is easily determined. All of the ingredients are known at the start (the spot price, interest rates, storage costs). Because this strategy generates the same payoff as the forward contract, the forward price can be determined from the cost of constructing the strategy today; otherwise, an arbitrage opportunity exists. All’s right with the world after all! The reasonable assumptions underlying the cost-of-carry model are frictionless markets, no cash flows to the underlying commodity, no counterparty credit risk, and competitive markets. Interest rates and commodity prices are random. Financial assets that satisfy the no cash flow assumption include stocks that pay no dividends over the forward’s life and zero-coupon Treasury bonds that mature after the delivery date. The final results of this chapter use the cash-and-carry argument to value a forward contract at an intermediate date and to establish a link between two different maturity forward prices on the same commodity. We adopt a data-model-arbitrage approach that is used throughout the book. This approach starts with a numerical example (data). Next, we replace numbers with symbols (model). Finally, we generate arbitrage profits when securities are mispriced (arbitrage).

11.2

A Cost-of-Carry Example

The cost-of-carry model uses the concept of no arbitrage. We develop the basic cost-of-carry model via either of two equivalent techniques, which we labeled the law of one price and nothing comes from nothing in chapter 6. These simple techniques rest on several key assumptions. We discuss the assumptions after illustrating the ideas involved.

A COST-OF-CARRY EXAMPLE

Example 11.1 uses the law of one price. It considers two different ways of buying a stock in the future: buying a forward contract or buying the stock in the spot market and carrying it to the future. We create these two positions so that we can extract the forward price by equating their values today. We replace numbers with symbols to obtain the basic cost-of-carry model.

EXAMPLE 11.1: Finding the Forward Price Using the Law of One Price

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The Data ■

Today is January 1. Suppose that we go to the forward market and buy a one-year forward contract on Your Beloved Machine Inc. (YBM) stock. Unless noted otherwise, all computations are on a per stock basis. No cash changes hands today because the forward price F is the fair price. At maturity, we receive one YBM stock worth S(T) from the forward’s seller by paying him the forward price F dollars. Record the cash flows (in Table 11.1a) as 0 today (time 0) and S(T) – F on the delivery date (time T). This is Portfolio A, which we label as the market traded forward.

■

Alternatively, one can buy one share of YBM in the spot market today and carry it to the future. Assuming that YBM pays no dividends over the forward’s life, the cash flows (in Table 11.1b) are –100 today and S(T) on December 31.

■

To create a future liability equal to the forward price, borrow its present value by shorting zerocoupon bonds. This cash inflow reduces today’s net payment. If the interest rate is fixed at 6 percent per year, the cash flows are F/(1 + interest) = F/1.06 today and –F a year later (see Table 11.1b). These long stock and short bond positions make up portfolio B, which is our synthetic forward.

■

By the law of one price, to prevent arbitrage, portfolios A and B with identical future payoffs must have the same value today. Consequently, −100 + F/1.06 = 0 or F = 100 × 1.06 = $106

(11.1)

The forward price equals the amount repaid on the loan for the stock purchase.

TABLE 11.1A: Portfolio A: Long Market Traded Forward Portfolio

Today (Time 0) Cash Flow

Delivery Date (Time T) Cash Flow

Long forward (forward price F)

0

S(T) – F

Net cash flow

0

S(T) – F

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TABLE 11.1B: Portfolio B: Long Stock and Short Bonds (Long Synthetic Forward) Portfolio

Buy YBM in the spot market (stock price S = $100) Short zero-coupon bonds to borrow the present value of the forward price F (interest rate is 6 percent per year) Net cash flow

Now (January 1) Cash Flow

Maturity Date (December 31) Cash Flow

–100

S(T)

F/1.06

–F

–100 + F/1.06

S(T) – F

The Model ■

Use symbols to generalize. Replace 100 with S and 0.06 with R to get our first result (stated later as Result 11.1): F = S (1 + R) (11.2a) or S = BF

(11.2b)

where (1 + R) is the future value of $1 invested today (which is also known as the dollar return) and B ≡ 1/(1 + R) is today’s price of a zero-coupon bond that pays $1 at maturity.

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A Graphical Approach ■

■

■

One can also develop the cost-of-carry model using payoff diagrams. Recall from chapter 5 that these diagrams plot “Payoffs” (or gross payoffs) along the vertical axis and the “stock price on the delivery date,” S(T), along the horizontal axis. First, buy one YBM stock. Its payoff is a straight line starting at the origin and rising at a 45∘ angle. If the stock is worthless on the forward’s delivery date, then you get 0; if it’s worth $5, then you get $5, and so on (see Figure 11.1). Next, draw another diagram under this depicting the payoff from a short bond position with a future liability of F dollars. This is a line parallel to the horizontal axis that begins at –F on the vertical axis. Now, vertically add up the payoffs from these two graphs. The short bond trade lowers the long stock’s payoff by a parallel downward shift of F dollars. Consequently, the combined payoff starts at –F along the vertical axis, rises at a 45∘ angle to the horizontal axis at F, and keeps increasing. The combined position is a synthetic forward because its payoff on the maturity date is identical to that of a market-traded forward contract. Because the forward price F is chosen so that the traded forward contract’s value is zero, to prevent arbitrage, the synthetic forward must also have zero value. Setting the synthetic forward’s cost of construction (S – BF) to zero yields the cost-of-carry model.

Arbitrage Profits ■

What would you do if the forward price differs from the arbitrage-free price? Suppose that an errant trader quotes a forward price of $110. As this is higher than $106, sell the overpriced forward, buy

A COST-OF-CARRY EXAMPLE

the stock, and borrow the present value of $110. Then – 100 + 11/1.06 = $3.77 is your immediate arbitrage profit—and you have no future cash flow because the short traded forward and the long synthetic forward’s cash flows exactly offset each other. If the trader quotes $100 instead, reverse the trades and make –100/1.06 + 100 = $5.66. Any forward price other than the arbitrage-free price of $106 creates an arbitrage opportunity.

FIGURE 11.1: Cost-of-Carry Model Gross payoff

Long stock + Short bond

Long stock

0 S(T), stock price on delivery date Payoff 0

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S(T) Payoff similar to long forward

Short bond

−F Payoff

0 F

S(T)

−F

This example illustrates that there is nothing magical about the forward price. The forward price is linked to today’s spot price via the cost-of-carry model, which only includes the interest cost in this simple setting. This simple model raises several questions: 1. What are the hidden assumptions underlying this argument? The next section discusses the assumptions underlying this and most other models in this book.

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2. Is the cost-of-carry model useful in other contexts? In its simple form, it generates the forward prices for investment assets like currencies and stocks. Forward prices for consumption or production assets like corn, copper, and oil, which are primarily held for consumption or as inputs to production, may be obtained by modifying this model to incorporate additional carrying charges such as transportation and storage costs as well as convenience yield benefits. This extension is presented in Chapter 12. The remainder of this chapter generalizes this example to build the cost-of-carry model. First, we need to introduce the hidden assumptions underlying our arguments.

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11.3

The Assumptions

A model is a simplified or idealized description, often expressed mathematically, that helps us understand and explain reality. Financial models serve this purpose. Being abstractions, models rely on simplifying assumptions. To understand a model, one needs to understand the assumptions. First, they show when to use, when not to use, and how not to abuse a model. Second, they give a sense of when the model fails and the biases that may result. Understanding the assumptions changes a model from a black box to an open box. We need the following assumptions for derivative pricing models in general and for forward pricing models in particular. A1. No market frictions. We assume that there are no market frictions like transaction costs (brokerage commissions and bid/ask spreads), margin requirements, short sale restrictions, taxes that may be levied at different rates, and assets that are not perfectly divisible. The frictionless market assumption helps us create a benchmark model against which other models can be compared. Once you understand this ideal setting, you can add back frictions to better capture reality. In addition, it’s a reasonable first approximation for many institutional traders. For example, professional traders at major investment banks have far lower transaction costs than do small investors. Professionals often trade within the bid/ask spread, they have less stringent margin requirements, they can avoid short sale restrictions, and their trading profits are taxed at a uniform rate. Furthermore, frictionless markets may now be a reasonable first approximation for small traders in some markets because of the accumulated advances in information technology that have drastically reduced trading costs. A2. No credit risk. Also known as default risk or counterparty risk, this is the risk that the counterparty to a transaction will fail to perform on an obligation. We assume that there is no credit risk. If credit risk exists, cash flows become less predictable, and we stumble at the onset. No credit risk is a reasonable approximation for exchange-traded derivatives, whose performance is guaranteed by a clearinghouse. Vigilant exchange officials, alert brokers and dealers, the customer’s margin account, and the exchange’s capital minimize this risk. Over-the-counter derivatives, however, may possess substantial credit risks. In practice, prudent actions, such as trading only with counterparties

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THE ASSUMPTIONS

that have high credit ratings, having well-documented legal contracts, and requiring parties to keep collateral, reduce this risk. Pricing derivatives in the presence of credit risk is an area of active research. Jarrow and Turnbull (1995) developed a popular model for pricing credit derivatives that is used in industry, known as the reduced form approach. This extension is discussed in Chapter 26. A3. Competitive and well-functioning markets. In a competitive market, traders’ purchases or sales have no impact on the market price; consequently, traders act as price takers. In a well-functioning market, price bubbles are absent. We assume competitive and well-functioning markets. In a competitive market, a trader is individually too small for his trades to have a quantity impact on the price and to possess market power. Consequently, he acts as a price taker vis-à-vis the market. The competitive markets assumption is the workhorse of modern economics, and many powerful results are derived from it. As illustrated via the futures market manipulations described in chapter 10, this assumption does not always hold. Pricing securities when traders have market power and when prices can be manipulated is a difficult exercise. One must take into account bargaining and strategic interactions—ideas and concepts that are outside the scope of this book. Consequently, the cost-of-carry model is a poor model in such a setting. Focusing on competitive markets avoids these complications. We also assume that there are no asset price bubbles. A price bubble happens when an asset’s price deviates from its intrinsic or fundamental value. The fundamental or intrinsic value can be defined in two equivalent ways. The first is the present value of its future cash flows. The second is the price one would pay if, after purchase, you had to hold the asset forever. If you reflect on these, it becomes clear that they are equivalent definitions. A difference between the market price and the fundamental value only occurs if one believes that selling (retrading) can generate a higher value than holding forever—this difference is the bubble. The terminology reflects the popular belief that such deviations are short-lived and surely burst, like a soap bubble. It has been shown that many Internet stocks exhibited a price bubble during the 1990s, when they reached astonishingly high values despite an absence of dividend payments and continued reports of zero or negative earnings. We rule out price bubbles because our pricing arguments fail to hold in this context. Pricing derivatives in noncompetitive markets and in the presence of bubbles is a daunting task and a hot area of research. We return to this topic again in Chapter 19, when we discuss pricing options. A4. No intermediate cash flows. As a first pass, we assume that the underlying commodity has no cash flows over the forward contract’s life. This is a simplification that we will need to relax. When considering stocks, most companies reward their shareholders by paying dividends. US companies tend to be fairly conservative about their dividend policies. They pay fixed dividends and dislike tinkering with either the payment date or the amount, lest the market infers something negative from their actions. In addition, many commodities receive income or cash flows—coupon-paying bonds or consumption/production assets that provide convenience yields are prime

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examples. Moreover, consumption/production assets tend to incur transaction, storage, insurance, and other costs. For the present, we ignore these costs. A5. No arbitrage opportunities. This assumption’s meaning is self-evident. These five assumptions form the bedrock of our analysis, and they determine the validity of our models. Talented traders regularly monitor the market and check whether the assumptions are satisfied. If the assumptions aren’t satisfied, the pricing models and trading strategies may fail to work. In such a situation, one should not rely on the model’s implications. If one is using the model to hedge, the hedge may no longer work. An optimal strategy may be to close out the position, rethink, and come back another day. As in Kenny Rogers’s song, “you got to know when to hold ’em, know when to fold ’em, know when to walk away, know when to run.”

11.4

The Cost-of-Carry Model

We now assemble the different components that we have developed to formally derive our forward pricing model. If the notations seem abstract, remember the data from our example (S is $100, 1 + R = 1/B = $1.06) to get a numerical feeling for the argument.

The Model Setup

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The cost-of-carry forward pricing model has the following structure: ■

There are two dates. At time 0 (which is today), we start the clock, trade, and create one or more portfolios of securities. The clock stops at some later time T.

■

Three securities trade: a stock, a forward on the stock, and a zero-coupon bond. Unless specified otherwise, we use the terms asset, commodity, stock, and spot somewhat interchangeably.

■

The stock price is S(0) ≡ S today and S(T) at time T. The forward contract starts today and a forward price F(0, T) ≡ F is determined in the market such that no cash changes hands. This is the fair price. To understand why this price is fair, suppose the contrary. If it were not fair, then one side of the contract would be entering into the contract with an immediate loss and with no additional compensation.

■

The contract ends on the delivery or maturity date by cash settlement or physical delivery. The long’s payoff is S(T) – F and the short’s payoff is –[S(T) – F], which are equal and opposite.

■

Traders can borrow or lend funds at the risk-free rate by shorting or buying zerocoupon bonds. Let B(0, T) ≡ B be today’s price of a zero-coupon bond that pays one dollar at time T. Let 1 + R be the dollar return obtained at time T from investing one dollar in the zero today.

We utilize the law of one price (from Chapter 6) to formally derive the cost-ofcarry model.

THE COST-OF-CARRY MODEL

Using the Law of One Price To find the arbitrage-free forward price F, let’s consider two different ways of buying the stock in the future without a cash flow today. First, buy a forward contract. This cashless trade obliges you to acquire the stock by paying the forward price on the delivery date (this is our Portfolio A). Second, buy the stock today and finance this purchase by borrowing. This costless transaction gives you the stock and a liability equal to today’s stock price with its accrued interest on the same maturity date (this is our Portfolio B). These transactions are summarized as follows: Portfolio A (buy forward): Zero-cost portfolio (today) → Get the stock, pay the forward price F (write this as S[T] – F ) (maturity date) Portfolio B (buy the stock, sell the zero-coupon bonds to finance the stock purchase): Zero-cost portfolio (today) → Get the stock, repay the loan (write this as S[T] – S[1 + R]) (maturity date) Note that these two final payoffs only differ by the constant [F – S(1 + R)] and that this constant is known today. These two portfolios with identical initial values (zero) must also have the same value on the delivery date (see Figure 11.1). If one of these values differs, it opens up an arbitrage opportunity that will be exploited by vigilant traders standing ready to pick up money lying on the street—they will trade until the two prices are equal and no arbitrage remains. This implies that the forward price must equal to today’s stock price repaid with accrued interest, that is, [F = S(1 + R)].

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Using Nothing Comes from Nothing Instead of using the law of one price, if we subtract portfolio B from portfolio A, we could use the nothing comes from nothing principle instead. This technique creates a new portfolio that has a zero value in the future and therefore must also have zero value today. We show this trading strategy in Table 11.2, which shows the trades and arbitrage opportunity. As depicted, this portfolio is costless to create and worth [S(1 + R) − F] on the delivery date. To prevent arbitrage, a portfolio that is worthless today with a constant value at delivery must also have zero value on the delivery date. This is the nothing comes from nothing principle. The trading strategy is called a reverse cash-and-carry because it involves short selling the asset and making good on this obligation in the future. Alternatively, we could have done a cash-and-carry where we purchase the spot and carry it to the future by paying the necessary carrying costs. We skip the details because you can readily obtain the result by reversing the trades and changing the signs in Table 11.2. Figure 11.2 graphically shows the reverse cash-and-carry argument.

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TABLE 11.2: Arbitrage Table for Reverse Cash-and-Carry Portfolio

Today (Time 0) Cash Flow

Maturity Date (Time T) Cash Flow

Buy forward

0

S(T) – F

Short sell stock

S

–S(T)

Invest proceeds from short selling by buying risk-free zero-coupon bonds

–S

S(1 + R)

Net cash flow

0

S(1 + R) – F

FIGURE 11.2: No-Arbitrage Chain for Cost-of-Carry Model

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Now (Cash flow)

Maturity (Cash flow)

Buy forward Cost 0

Payoff S(T) – F How to get rid of S(T)?

Short sell stock Proceeds (+S)

Liability – S(T) Asset disappears

Lend proceeds Buy zero-coupon bonds worth S

Payoff S(1 + R)

Zero value (for sure)

The constant net payoff [S(1 + R) – F] must be zero (otherwise an arbitrage opportunity)

Different Methods for Computing Interest Remember from chapter 2 that we can compute B and R by using various discrete or continuous compounding methods. So far, we have computed B by using simple interest, which is the easiest case. We can also use continuous compounding by writing the zero-coupon bond price B or dollar return 1 + R as B = e−rT and 1 + R = 1/B = erT where r is the continuously compounded interest rate.

THE COST-OF-CARRY MODEL

We gather all our results together.

RESULT 11.1 Cost-of-Carry Model Consider a forward contract on a commodity beginning at time 0 (today) and maturing at time T. Then F = S (1 + R) (11.2a) Equivalently, S = BF

(11.2b)

where F ≡ F(0, T) is today’s forward price for delivery at time T, S ≡ S(0) is today’s spot price of the underlying commodity, 1 + R is the dollar return at time T from investing $1 in a zero-coupon bond today, and B ≡ 1/(1 + R) is today’s price for a zero-coupon bond that pays $1 at time T. Dollar returns are computed by the formulas Simple interest ∶ 1 + R = (1 + i × T) Continuously compounded interest ∶ 1 + R = erT

(11.2c) (11.2d)

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where i is the simple interest rate per year and r is the continuously compounded interest rate per year.

Continuously compounded interest rates give a better approximation than do simple interest rates when compounding is computed daily. Moreover, we will show in the next chapter that this assumption lays the foundation for developing a variety of cost-of-carry models, including those written on indexes and commodities that incur storage costs and have convenience yields. Example 11.2 uses this formula to compute forward prices and capture arbitrage profits.

EXAMPLE 11.2: Finding Forward Prices and Capturing Arbitrage Profits

The Forward Price ■

Consider the data from Example 11.1. The YBM stock price S is $100 today, a newly written forward on YBM matures in one year, and the simple interest rate is 6 percent per year. Then Equation 11.2c gives Dollar return 1 + R = (1 + 0.06 × 1) = $1.06 and the zero-coupon bond price B = 1/1.06 = $0.9434

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By expression (11.2a), the forward price is F = S (1 + R) = $106.00 which is the same price as in Example 11.1.1

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Capturing Arbitrage Profits ■

Suppose that an errant trader quotes a forward price of $110. Then the spot and forward prices are too far apart. The spot is $100, while the discounted forward price BF = $103.77 so that S – BF = – $3.77.

■

One can profit by doing a reverse cash-and-carry arbitrage. Simultaneously buy the relatively underpriced stock, sell the relatively overpriced forward, and short zero-coupon bonds with a maturity value of $110 to balance the portfolio. Immediately collect $3.77. The net payoff will be zero for sure on the maturity date.

■

Want to take your profits at maturity? Modify the bond position—short only enough zero-coupon bonds to finance the spot purchase, and make F – S(1 + R) = $4 on December 31.

■

Now, consider a different scenario—use the same data as before, except let the forward price be $100. Here S – BF = $5.66, suggesting that either the stock price is too high or the forward price is too low (or both). Do a reverse cash-and-carry arbitrage by selling the stock, buying the forward, and buying zero-coupon bonds. As before, the bond position can be tinkered with to take the profit now or later. If you buy zero-coupon bonds with a face value of $100, then you will make –BF + S = $5.66 of arbitrage profits today.

1 If this is continuously compounded interest, r is 6 percent, and then by (11.2a) and (11.2d), the forward price is F = S(1 + R) = 100 exp(0.06 × 1) = $106.18.

The Arbitrage Table Approach The previous examples motivate an arbitrage table approach for solving problems and proving results. By systematically recording cash flows at different dates, an arbitrage table enables direct comparison, which makes proofs easier. Although there are no set rules, we suggest three possible methods: 1. Understand the problem, using numbers if necessary. What do the different variables mean? Which security generates what cash flows? When? Draw a timeline. Plug dates and variable values into the formula to get a feel for the result to prove or how to make arbitrage profits. Use your intuition to sense which variables are overvalued or undervalued, using our “buy low, sell high” dictum. 2. Interpret cash flows. If the cash flows are positive today and zero on the delivery date, then you have made arbitrage profits. Alternatively, if cash flows are zero today and have a positive value on the delivery date, then you also have made arbitrage profits.

VALUING A FORWARD CONTRACT AT INTERMEDIATE DATES

3. Gather variables to one side of the equality and set the net cash flows at one of the dates to zero. It is inconvenient to work with nonzero cash flows at both the starting and ending dates. Life is easier if you gather variables to one side of an equality so that you can set today’s or the delivery date’s net cash flows to zero. Then, to prevent arbitrage, the other net cash flow must also be zero.

11.5

Valuing a Forward Contract at Intermediate Dates

Suppose you bought a forward contract some time back, but it still has remaining life. How much it is worth today? If the spot price soars or plunges, how would it affect the forward’s value? We use a cash-and-carry argument to develop a formula that answers these questions. The model may be used for mark-to-model accounting, which requires the valuation of assets and liabilities on a daily basis, a practice that is increasingly becoming popular with companies and regulators. Example 11.3 introduces a numerical example that is then generalized to obtain the relevant result.

EXAMPLE 11.3: Valuing a Forward Contract at an Intermediate Date

The Data

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■

Suppose that you bought a forward on January 1. Three months have passed. Meanwhile, the underlying price has increased. Of course, your long position has gained value because you entered into a contract to buy the underlying at a fixed price, but the underlying has become more valuable. What is the exact value of your long position today? Let us consider an example that uses some of our previous data: - On January 1 (time 0), YBM’s stock price S(0) was $100. - The simple interest rate i was 6 percent per year. - You purchased a newly written forward contract on YBM that matures in a year on December 31 (time T). - The forward price was set at F(0) = S(1 + R) = $106.

■

Three months have passed; today is April 1 (time t). YBM has rallied and the new stock price S(t) ≡ S is $120. You are still obliged to buy YBM by paying F(0) in nine more months (T – t is 0.75 years). But what about the value of your forward, V (t) ≡ V ? If you sell your position, how much should you get paid?

■

Draw a timeline to keep track of the cash flows (see Figure 11.3). The stock price increase has made the long forward position more valuable, the exact amount of which is determined by the arbitrage table (see Table 11.3).

■

Begin the portfolio construction as if you are buying the forward position by paying V. This has a payoff of S(T) – F(0) = S(T) – 106 on December 31. Next, short sell the stock. This gives $120 today but creates a liability –S(T) on the delivery date. The stock disappears, but you still have to come up

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with $106. This can be arranged by lending the present value of $106 via buying zero-coupon bonds. Notice that on April 1, the price of a zero-coupon bond that pays $1 on December 31 is B ≡ B (t, T) = 1/ (1 + 0.06 × 0.75) = $0.9569 ■

The portfolio has a zero value on December 31. To prevent arbitrage, −V + 120 − 0.9569 × 106 = 0 or V = $18.561

(11.3)

The Model ■

Replace numbers with symbols in Equation 11.3 to get −V + S − BF (0) = 0 or V = S − BF (0)

1

Instead, if we assume a continuously compounded interest rate r of 6 percent per year fixed, then F(0) = S(1 + R) = 100 × exp(0.06 ×exp(0.06 × 1) = $106.18, B(t, T) = exp(– 0.06 × 0.75) = $0.9560, and V = 120 – 0.9560 × 106.18 = $18.49. There is a 7 cent difference in value of a forward contract because of the different ways of computing interest.

This is Result 11.2, which is formally stated as follows.

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RESULT 11.2 Valuing a Forward Contract at an Intermediate Date Consider a forward contract created at time 0 and that matures at time T. On an intermediate date t (today), V = S − BF (0)

(11.4a)

where F(0) is the forward price at time 0 for a contract maturing at time T, V ≡ V (t) is today’s value of a long forward position, S ≡ S(t) is today’s spot price of the underlying commodity, and B ≡ B(t, T) is today’s (time t) price of a zero-coupon bond that pays one dollar at time T. In the case of simple interest, B is 1/[1 + i(T – t)], where i is the interest rate per year. In the case of continuously compounded interest, B is e–r(T − t) where r is the interest rate per year.

This is a more general version of our first result. If we set V = 0, as happens on a forward contract’s starting date, then Result 11.1 becomes a special case of Result 11.2. Using Results 11.1 and 11.2 together, we can obtain another insight. First,

VALUING A FORWARD CONTRACT AT INTERMEDIATE DATES

TABLE 11.3: Valuing a Forward Contract That Began Earlier Portfolio

Today, April 1 (Time t) Cash Flow

Buy forward (worth V (t) or V to long), forward price F(0) = $106

–V

Short sell spot (S = 120) Lend present value of forward price BF(0) Net cash flow

Delivery Date, December 31 (Time T) Cash Flow

S(T) – 106

120

–S(T)

–(1/1.045)106

106

–V + 120 – 0.9569 × 106

0

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FIGURE 11.3: Timeline for a Forward Contract That Began Earlier

Time 0

Forward started on January 1.

Time t (today)

Time T

B = 0.9569

$1

Today (April 1).

Forward matures one year from starting date.

Forward price Value of forward contract to long is V(t). Value of forward to long is S(T) – F(0). F(0) negotiated. Value of forward to long is 0.

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we note that Result 11.1, when applied to the time t forward price, implies that S = BF(t). Second, substitution of this fact into expression (11.4a) and some simple algebra yields an alternative expression for the value of the forward contract at time t: V = B [F (t) − F (0)]

(11.4b)

This expression shows that the value of the forward contract increases by the present value of the change in the new delivery price, F(t), relative to the original delivery price, F(0). The present value is needed to reflect the fact that the payments are not made until the delivery date, time T.

11.6

Linking Forward Prices of Different Maturities

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Consider a market in which the cost-of-carry model determines forward prices with high accuracy. If we change the delivery date, the model gives us a series of forward prices corresponding to contracts maturing on those different dates. We can also use the cash-and-carry argument to develop a formula linking these prices. This enables us to find one forward price when only the carrying cost and another forward price are known. When several forward contracts exist on the same commodity, this model also enables us to determine whether all the forward prices are correct or some of them are out of line. The model is given as our Result 11.3. A simple proof follows the result.

RESULT 11.3 Link between Near- and Distant-Maturity Forward Prices Consider two forward contracts that begin at time 0 (today), mature on the near-maturity date (time n) and the distant-maturity date (time d), respectively. Then, standing at time n, F (0, n) =

B (0, d) × F (0, d) B (0, n)

(11.5)

where F(0, n) and F(0, d) are the forward prices negotiated at time 0 for forward contracts maturing at times n and d, respectively, and B(0, t) is the price of a zero-coupon bond at time 0 that pays $1 at times t = n and t = d. The zero-coupon bond price B is 1/(1 + it) for simple interest and e–rt for continuously compounded interest, where i and r are the respective annual interest rates.

SUMMARY

Although we view this result as an application, it’s also a generalization. If you set the near-maturity forward to be at its delivery date n, then by convergence of the basis, F(0, n) becomes the spot price S(n), B(n, n) = 1, and expression (11.5) gives back Result 11.1 at time n. To prove Result 11.3, we just use Result 11.1 twice, once for time n and once for time d, getting the two equations S (0) = B (0, d) × F (0, d) S (0) = B (0, n) × F (0, n) Setting these two equations equal and solving for F(0, n) gives the result.

11.7

Summary

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1. We use no arbitrage via the law of one price and the principle that nothing comes from nothing introduced in chapter 6 to develop a forward pricing model. First, we create a securities portfolio to replicate a market-traded forward’s payoff. To avoid arbitrage, the cost of the replicating portfolio must equal to the forward contract’s arbitrage-free value. Equivalently, an alternative portfolio going long the forward contract and selling short the replicating portfolio (or vice versa) will have a zero value both now and in the future. Both approaches give the same price. 2. A portfolio consisting of a long stock, a short forward, and short some zero-coupon bonds can be created so that it has zero value on the forward’s maturity date. To prevent arbitrage, it must also have a zero value on the starting date. The forward price is determined from this condition. This gives Result 11.1, which is the basic cost-of-carry model. It states that the forward price is the future value of the spot price or F = S(1 + R). 3. The cost-of-carry model makes a number of assumptions: no market frictions, no credit risk, competitive and well-functioning markets, no dividends over the contract’s life, and no arbitrage opportunities. 4. There are several ways of developing cost-of-carry models and solving arbitrage problems. We primarily use arbitrage tables, which systematically record cash flows at different dates. 5. We use the cash-and-carry argument to build more complex models. Result 11.2 states that the value of a forward contract at an intermediate date equals the spot price minus the discounted value of the old forward price or V = S – BF(0). Result 11.3 states that when two different forward contracts are written on the same commodity but with different maturity dates, the near-maturity forward price is a fraction of the distant-maturity forward price or F(0, n) = [B(0, d)/ B(0, n)] × F(0, d).

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11.8

Cases

American Barrick Resources Corp.: Managing Gold Price Risk (Harvard Busi-

ness School Case 293128-PDF-ENG). The case discusses derivative usage for hedging by a gold mining company. Banque Paribas: Paribas Derives Garantis (Harvard Business School Case

295008-PDF-ENG). The case explores issues connected with a broker-dealer setting up a derivatives subsidiary to achieve a credit rating. Hedging Currency Risks at AIFS (Harvard Business School Case 205026-PDF-

ENG). In this case a company considers managing foreign currency risks with derivatives.

11.9

Questions and Problems

11.1. Today’s price of gold in the spot market is $1,500 per ounce. The price of a

zero-coupon bond maturing in one year is $0.95. What will be the one-year forward price for gold? 11.2. The current price of Your Beloved Machine’s stock is $109. The continuously

compounded interest rate is 5.25 percent per year. What will be the fivemonth forward price for YBM stock? 11.3. The spot price of silver is $30 per ounce. The simple interest rate is 6 percent Copyright © 2018. World Scientific Publishing Company Pte. Limited. All rights reserved.

per year. The quoted six-month forward price for silver is $31. a. What should be the arbitrage-free forward price for silver for a forward

contract maturing in six months? b. Demonstrate how you can make arbitrage profits in this market. 11.4. Today’s spot price of silver is $30 per ounce. The simple interest rate is

6 percent per year. The quoted six-month forward price for silver is $32. Transaction costs are $0.10 per ounce whenever spot silver is traded and a $0.25 per ounce one-time fee for trading forward contracts but no charges for trading bonds. a. Demonstrate how you can make arbitrage profits in this market if there are

no transaction costs. b. If you have to pay transaction costs, demonstrate how you will make

arbitrage profits or explain why you cannot make any such profits. 11.5. Suppose that the continuously compounded interest rate is 6 percent per year,

and the nine-month forward price for platinum is $1,750. What is today’s spot price of platinum? 11.6. What does the assumption of no market frictions mean? Is this assumption

true in current commodity markets?

QUESTIONS AND PROBLEMS

11.7. If the assumption of no market frictions is not true, then why should we study

models using this assumption? 11.8. What is the competitive market assumption, and what adverse market

behavior does it exclude? 11.9. What are the five assumptions underlying the cost-of-carry model for pricing

forward contracts? Which of these assumptions are most likely to be satisfied in current commodity markets? 11.10. State the key result of the cost-of-carry model in your own words. 11.11. Today is January 1. Forward prices for gold forward maturing on April 1 is

$1,500 per ounce. The simple interest rate is 6 percent per year. What would be the forward price for a forward contract on gold maturing on August 1? 11.12. Today is January 1. Forward prices for contracts maturing on April 1 and

on October 1 are $103 and $109, respectively. The simple interest rate is 8 percent per year. Assuming the spot price is $100 today, demonstrate two ways in which you can make arbitrage profits from these prices. 11.13. Today is January 1. Forward prices for contracts maturing on April 1 and

on October 1 are $103 and $109, respectively. On April 1, the price of a zero-coupon bond maturing on October 1 is $0.97. Assuming that the underlying interest rate is a constant interest rate, demonstrate one way of making arbitrage profits from these prices. 11.14. State the equation for the valuation of a forward contract in your own words.

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11.15. Suppose you bought a forward on January 1 that matures a year later. The

forward price was $214 at that time, and the simple interest rate was 7 percent per year. Six months have passed, and the spot price is now $150. What is the value of your forward contract today? 11.16. Suppose you trade a forward contract today that matures after one year. The

forward price is $105, and the simple interest rate is 7 percent per year. If, after six months from today, the spot price is going to be $150 and the value of the forward contract is $20, demonstrate how you can make arbitrage profit from these prices. 11.17. The value of a forward contract that you have been holding for the last six

months is $50 today. It matures in 3 more months. If today’s spot price is $100 and the underlying interest rate is 5 percent, what was the forward price that you had negotiated when you purchased the contract six months back? 11.18. Explain the difference between the forward price and the value of a forward

contract. How are they related? 11.19. What is the relation between forward prices of different maturities on the

same underlying? 11.20. (Excel) For gold and silver, collect from the internet today’s spot prices as well

as futures prices for futures contracts maturing up to one year. Also collect Treasury bill prices of different maturities. Using the above data, perform the

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following for three of the most actively traded futures contracts for gold as well as for silver: a. Assuming that interest is the only cost of carry, use Result 11.1 (Cost-of-

Carry Model) of Chapter 11 to compute forward prices that correspond to the maturities of three of the most actively traded contracts for the two metals. b. Compare these prices to the traded futures prices. Do they seem signifi-

cantly different? If so, explain why.

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Hints and Suggestions: ■

New York spot price for gold and silver can be found on a metal dealer’s (such as Kitco’s) website.

■

Futures prices can be obtained from an exchange’s (such as the CME Group’s) website www.cmegroup.com/market-data/index.html

■

Daily Treasury bill rates data can be found from the US Treasury’s website www.treasury.gov/resource-center/data-chart-center/interestrates/Pages/TextView.aspx?data=billrates

■

Use settlement prices wherever possible. If ask and bid prices are quoted for the spot or futures, take the average of the two.

■

Exclude futures contracts maturing in the current month as they can have liquidity problems.

12 The Extended Cost-of-Carry Model Copyright © 2018. World Scientific Publishing Company Pte. Limited. All rights reserved.

12.1 Introduction 12.2 A Family of Forward Pricing Models 12.3 Forwards on Dividend-Paying Stocks The Cost-of-Carry Model with Dollar Dividends

12.4 Extended Cost-of-Carry Models 12.5 Backwardation, Contango, Normal Backwardation, and Normal Contango 12.6 Market Imperfections 12.7 Summary

The Cost-of-Carry Model with a Dividend Yield

12.8 Cases

A Synthetic Index

12.9 Questions and Problems

The Foreign Currency Forward Price

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12.1

Introduction

Cornell University is situated in rural upstate New York. The university campus is built on a hill overlooking beautiful Lake Cayuga and the city of Ithaca. The landscape is scenic, dotted with hills, valleys, and even a few gorges formed during the ice age. The winters are cold, snowy, and icy, which makes driving the roads treacherous. When choosing a car to own in Ithaca, a resident has many options to consider. Key among these options is the choice of rear-wheel, front-wheel, or four-wheel drive. To safely navigate the wintry roads, four-wheel drive is the preferred selection. Consequently, four-wheel drive cars are in great demand. Analogous to an Ithacan deciding which car to buy in hilly Ithaca, when constructing a forward pricing model, one needs to consider the market setting carefully. Commodities differ in whether they have cash flows, storage costs, or convenience yields. Choosing the correct model for proper pricing and hedging requires matching the model’s assumptions with market conditions. To do so otherwise leads to “treacherous driving” when using the model in practice. Chapter 11 presented the arbitrage-free forward pricing model under the assumption that the underlying commodity has no cash flows, storage costs, or convenience yields over the forward contract’s life. However, stocks and stock indexes pay dividends, bonds make coupon payments, foreign exchange prices are affected by both domestic and foreign interest rates, and physical commodities like corn or gold incur storage costs but may receive a convenience yield. The model in Chapter 11 also assumed frictionless markets and equal borrowing and lending rates, but trading involves brokerage commissions as well as the bid/ask spread, and borrowing and lending rates differ. How does one modify the forward pricing model when these assumptions are relaxed to get a better approximation? The purpose of this chapter is to answer this question. We do this by relaxing assumptions A1 and A4 of Chapter 11 and thereby incorporate dividends, storage costs, convenience yields, and market imperfections into our framework. Although this additional complexity appears daunting, it isn’t. The basic logic of the cost-of-carry model still applies, but with bells and whistles attached. Most of these extensions can be easily handled with only minor modifications to the basic model. By the end of this chapter, one should be able to choose the safest “vehicle” for forward pricing in realistic market “landscapes.”

12.2

A Family of Forward Pricing Models

The last chapter’s cost-of-carry model includes interest as the only cost-of-carry and invokes a cash-and-carry argument to express the forward price as the future value of the spot price (Result 11.1). As in Chapter 11, unless specified otherwise, we use the terms asset, commodity, stock, and spot interchangeably. Spot trading and a cash market transaction also have the same meaning. The cost-of-carry model relied on several critical assumptions that we repeat here for convenience: A1. No market frictions A2. No credit risk

A FAMILY OF FORWARD PRICING MODELS

A3. Competitive and well-functioning markets A4. No intermediate cash flows A5. No arbitrage opportunities This chapter studies the determination of forward prices in a variety of settings. Figure 12.1 provides a road map for the family of forward pricing models considered. We first generalize the model by relaxing the fourth assumption with the introduction of dividends. This gives the first three extended models of this chapter: ■

The simplest dividend model assumes that the underlying stock pays fixed dollar dividends on known dates (Result 12.1a). This model is useful for finding forward prices for contracts written on equities and coupon-bearing Treasury bonds that mature after the delivery date.

■

The second dividend model assumes that the underlying stock pays continuous dividends (Result 12.1b). It’s helpful for finding forward prices when the underlying is a stock index.

■

An important application of the second model is to determine forward prices for foreign currency forward contracts (Result 12.2).

FIGURE 12.1: A Family of Forward Pricing Models

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Cost-of-carry model (the basic model)

Link between forward prices of different maturities (an application)

Valuing a forward at an intermediate date With dividends

Currency forward (an application)

With storage costs and convenience yields

Generalized cost-of-carry

Backwardation Forward Price < Spot Price

Contango Forward Price > Spot Price

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The dividend model paves the way for richer models. One such extension is storage costs for holding a commodity mainly used for consumption or production like copper, corn, or crude oil. Storing the physical commodity may also provide a convenience yield like the ability to keep a production process running in times of temporary shortages. We include these costs and benefits to build further extensions of the model (Results 12.3 and 12.4). Last, we introduce market frictions such as brokerage costs, differential borrowing and lending rates, and limited access to proceeds from short sales. Here buys and sells are no longer symmetric. Well-defined prices give way to ranges within which arbitrage-free forward prices must lie (Result 12.5).

12.3

Forwards on Dividend-Paying Stocks

Most mature US companies pay their shareholders regular dividends (see Chapter 3 for a discussion of dividends). Companies dislike tinkering with the payment date or the dividend amount to minimize the possibility that markets will ascribe negative reasons for these actions and penalize the stock price. Firms like to wear the face of an English monarch from which you can read nothing. Consequently, fixed payment dates and fixed dollar dividends are reasonable assumptions for mature companies. Many assets, such as Treasury notes and bonds, have highly predictable cash flows and may be similarly modeled.

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The Cost-of-Carry Model with Dollar Dividends Consider a company that pays a fixed dollar dividend on a future date decided in advance. A dividend payment actually involves several dates including the declaration date and the report date. For our purposes, we only need to focus on the ex-dividend date, which is the cut-off for buying the stock with the dividend. Recall our discussion from Chapter 3. ■

Buying the stock before it goes ex-dividend gives the share plus the dividend; purchasing after this deadline gives the share without the dividend.

■

On the ex-dividend date, the cum-dividend stock price equals the ex-dividend stock price plus the dividend (see Result 3.1).

Dividends create an extra cash flow from holding the stock. If one adjusts the stock price for this cash flow, the standard argument follows. Example 12.1 shows how to do this. The resulting pricing model is identical to our cost-of-carry model, except that the stock price net the present value of the dividend replaces the stock price in the basic model.

FORWARDS ON DIVIDEND-PAYING STOCKS

EXAMPLE 12.1: The Forward Price when the Underlying Pays a Fixed Dollar Dividend

The Data ■

Consider the data from Example 11.1. Your Beloved Machine Inc.’s (YBM) stock price S is $100 today (January 1, time 0), and the simple interest rate is 6 percent per year. Consider a newly written forward contract on YBM that matures in one year (December 31, time T).

■

Now, assume that the stock pays a dividend div of $1 in three months (April 1, time t1 ). Figure 12.2 gives the timeline for these cash flows. Table 12.1 accommodates this dividend by adding an extra column to our arbitrage table.

■

To determine the forward price, we use the nothing comes from nothing arbitrage principle. We create a portfolio that has zero net payoffs on all future dates. To handle the dividend, we need to borrow cash to generate a future liability just equal to the dividend payoff. Here are the details: - Buy one share of YBM. This gives a dividend div = $1 after three months and a stock worth S(T) after one year. - Zap the dangling dividend by borrowing its present value. As B1 = 1/(1 + 0.06 × 0.25) = $0.9852 is the price of a zero-coupon bond maturing in three months, record the cash flows as B1 div = $0.9852 today and –$1 on the ex-dividend date. - Sell YBM forward to eliminate it from the portfolio on the maturity date. - Finally, get rid of the forward price F by borrowing its present value. Jot down the cash flows as BF = (1/1.06)F today and –F after one year.

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■

To prevent arbitrage, today’s net cash flow must also be zero: −100 + 0.9852 × 1 + 0.9434F = 0 or, F = $104.96

(12.1)

The Model and Arbitrage Profits ■

Replace numbers with symbols in (12.1) and rearrange terms to get S − B1 div = BF

■

(12.2)

Suppose a dealer forgets to adjust for the dividend and quotes a forward price of $106. To obtain arbitrage profits, create the portfolio given in Table 12.1. This has a short position in the overpriced dealer-quoted forward and a long position in the synthetic forward. Then −S + B1 div + BF = −100 + 0.9852 + (106/1.06) = $0.9852 The present value of the dividend is the immediate arbitrage profit.

■

If the quoted forward price is less than the arbitrage-free price, reverse the trades to capture the arbitrage profit.

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Formalizing this example gives our first result.

RESULT 12.1A Cost-of-Carry Model with Dividends Consider a forward contract written today on a stock that pays a fixed dollar dividend div on a known future date. Then S − B1 div = BF

(12.2)

where today (time 0) ≤ ex-dividend date (time t1 ) ≤ maturity date (time T), S is today’s spot price, F is today’s forward price for time T, and B1 ≡ B(0, t1 ) and B ≡ B(0, T) are today’s prices of zero-coupon bonds maturing on the ex-dividend and the delivery dates, respectively. The zero-coupon bond price B is 1/(1 + iT) in the case of simple interest and e−rT for continuously compounded interest, where i and r are the annual interest rates. B1 is computed by replacing T with t1 in these expressions.

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FIGURE 12.2: Timeline of Cash Flows for a Dividend Paying Stock

January 1 Time 0 (Today) B1 = 1/1.015 B = 1/1.06 Forward starts.

December 31 Time T (One year later)

April 1 Time t1 (Three months later) 1 1 YBM pays dividend div of $1 to the stockholder.

Forward matures. Long’s payoff is S(T) – F.

FORWARDS ON DIVIDEND-PAYING STOCKS

TABLE 12.1: Arbitrage Table for Finding the Forward Price of a DividendPaying Stock Portfolio

Buy stock, get dividend div after three months Borrow present value of the dividend B1 div today Sell forward Borrow present value of forward price BF Net cash flow

Today, January 1 (Time 0) Cash Flow

–100

(1/1.015) × 1

Dividend Date, April 1 (Time t1 ) Cash Flow

Delivery Date, December 31 (Time T) Cash Flow

1

S(T)

–1

0

–[S(T) – F]

(1/1.06)F

–100 + 0.9852 × 1 + 0.9434F

–F

0

0

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We can easily extend this to accommodate more dividends. Suppose that the stock pays another dividend div2 at time t2 before the forward matures. Then, the result gets modified to S − (B1 div1 + B2 div2 ) = BF (12.3) where div1 has a subscript to link it with the payment date and B2 is today’s price of a zero maturing on the second ex-dividend date. In its most general form, S − (Present value of all dividends over the forward’s life) = BF

(12.4)

Why adjust for dividends? We do these adjustments because dividends differently impact the stock and the forward: the long stock earns dividends but the long forward doesn’t. Once you take out the dividends accruing to the stock, it puts both positions on the same footing, and our familiar cost-of-carry argument applies. Next, we introduce forwards on indexes. For these contracts, the dividend adjustment is more complex.

The Cost-of-Carry Model with a Dividend Yield Modern portfolio theory (MPT; say it quickly and it comes out as “empty!”) pioneered the idea of investing in diversified portfolios that replicate the returns on stock market indexes. An index mutual fund (or a stock index fund) pursues this objective by holding a basket of securities in the same proportion as the index. The fund’s share price is usually determined at day’s end. An exchange traded fund (ETF) strives to do the same, but it trades continuously on an exchange.1 Indexes and index funds underlie a large number of derivative securities, including forward contracts.

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What is the arbitrage-free forward price of such an indexed portfolio? Unfortunately, the dividend adjustment method outlined in the previous section, which assumes that the dividends and their payment dates are nonrandom, works poorly for an index because there are too many dividends. This section, therefore, develops an alternative approach that accommodates random dividends and random payment dates. This modification uses dividend yields and the continuously compounded version of the cost-of-carry model (see Chapters 3 and 11, respectively). To apply this modification, we first need to understand how to adjust an index’s value for continuously paid dividends. And this, in turn, requires a discussion of synthetic index construction.

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A Synthetic Index An index is a portfolio of stocks (or assets) formed according to some rules. A synthetic index is a portfolio created for the purpose of replicating the returns on a traded index. Indexes come in two varieties: those that adjust for dividends and those that do not. Total return indexes assume that all disbursements from the portfolio’s stocks, including regular dividends, get reinvested in the index portfolio. For total return indexes, the reinvestment assumption effectively removes the dividends from consideration because there are no cash flows (see Figure 12.3). In this case, forwards on total return indexes can be priced using the no cash flow assumption, and the results of Chapter 11 apply. However, most indexes, including the Dow Jones Company’s and Standard and Poor’s, make no adjustment for regular cash dividends. In this case, dividends need to be explicitly considered. The trick is to compute the value of the index today (time 0), excluding the present value of the dividends paid over the life of the forward contract. To compute this adjustment, it is helpful first to consider how such an indexe’s value changes if all dividends are reinvested, as in a total return index. Let the value of the index today without reinvestment be S(0) and its value at time T be denoted S(T). Assume that it pays a continuous dividend yield of 𝛿 percent per year. For comparison purposes, consider an otherwise identical index with the same time 0 value I(0) = S(0). For this index, however, assume that all of the continuously paid dividends are reinvested back into this index. Let us call this the synthetic total return index. Because the dividends are reinvested into I(T), its value at time T will exceed that for the index where dividends are not reinvested, that is, I(T) > S(T). In fact, using Result 3.2, we know that I (T) = S (T) eᄕT Note that S(0) represents the present value of the index at time T plus the dividends. To get the present value of the index, less the continuously paid dividends over the life of the forward contract, we just remove the dividend yield term; that is, present value of S (T) = S (0) e−ᄕT

FORWARDS ON DIVIDEND-PAYING STOCKS

This expression gives us the modified “stock price” to use in the forward pricing model. Notice that this is a quantity adjustment in contrast to the price adjustment used previously for dollar dividends. This modification assumes that the dividends on the index are paid over the year at a uniform rate (see Figure 12.3). Although some dividends are smaller, while others are larger, this is a reasonable assumption if there are sufficiently many stocks in the index and dividends are paid randomly over time. Newspapers and other information vendors regularly report these dividend yields. Example 12.2 uses this stock price modification to determine the forward price on an index. In practice, an index derivative usually has a multiplier, for example, the Chicago Mercantile Exchange’s S&P 500 futures is $250 times the S&P 500 index level, and if the previous day’s futures price is 1,000 and today’s price is 1,000.10, then the long would have 250(1,000.10 – 1,000) = $25 credited to her brokerage account. As we have done before, we ignore the multiplier and do calculations on a per unit basis.

EXAMPLE 12.2: The Forward Price for an Index with a Dividend Yield

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The Data ■

Consider a fictitious “INDY index” obtained by averaging stock prices and a synthetic index “INDY spot” that replicates its performance. INDY’s current level S is 1,000, and the index does not reinvest dividends. A newly written forward contract on the index matures in a year (time T). Let the continuously compounded interest rate r be 6 percent per year.

■

Stocks constituting INDY index paid $20 of dividends last year and are expected to pay the same this year. Then, the dividend yield 𝛿 = 20/1,000 = 0.02 or 2 percent per year.

■

■

If you buy e−𝛿T = e−0.02×1 = 0.9802 units of INDY spot and reinvest the dividends on a continuous basis, then you will have one unit of the index worth S(T) at year’s end. Table 12.2 records the following trades in an arbitrage table: - Sell the forward contract on the INDY index. This gives a zero value at time 0 and –[S(T) – F] on the delivery date T. - Buy 0.9802 units of INDY spot, and record the cash flows as – 0.9802 × 1,000 = –980.20 today. Reinvest all dividends until delivery back into the index to get S(T) on the maturity date. - Borrow the present value of F. A zero-coupon bond maturing one year later is worth B = e−rT = e−0.06 = $0.9418 today. Record the cash flows as BF = 0.9418F on January 1 and –F on December 31.

■

The resulting portfolio has a zero payoff on the delivery date. To prevent arbitrage, the net payoff is zero today. Consequently, −0.9802 × 1, 000 + 0.9418F = 0 or F = $1, 040.81

(12.5)

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The Model and Arbitrage Profits ■

Replace the numbers with symbols in Equation 12.5 to get −e−𝛿T S + e−rT F = 0 or F = Se(r − 𝛿)T

■

Suppose an errant dealer quotes $1,020 as the forward price for INDY index. As this is less than the arbitrage-free price, buy the relatively underpriced traded forward and sell short the synthetic forward as in Table 12.2. This gives an immediate arbitrage profit of −e−rT F + e−𝛿T S = −0.9418 × 1, 020 + 0.9802 × 1, 000 = $19.60

FIGURE 12.3: Creating a Synthetic Total Return Index Today (Time 0)

Maturity (Time T)

(a) Total return index

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I(0)

…

…

I(T)

All dividends reinvested in the index. (b) Most indexes S(0)

…

…

S(T)

Underlying stocks pay numerous dividends, but the index is not adjusted. Now (Time 0)

End Date (Time T)

(c) Synthetic total return index S(0)

…

…

S(T)eδT

Reinvest all dividends as paid S(T) e−δT Spot)

Forward Price

Backwardation (Forward price < Spot)

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FIGURE 12.6: Normal Contango and Normal Backwardation Normal Contango

Forward Price Speculators (Net)

5.05

Hedgers (Net)

5 Short Contracts

–500 0

500

Long Contracts

Expected spot price is $5 Forward price is $5.05 in equilibrium (net demand = net supply)

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Normal Backwardation Farmer

Forward Price

Hedgers (Net)

5 4.95

Short Contracts

–450 0 450

Expected spot price is $5 Forward price is $4.95 in equilibrium

Speculators (Net) Long Contracts

MARKET IMPERFECTIONS

Consider another scenario where corn farmers, who want to hedge by selling forward and fix a selling price for their corn, dominate the forward market (see Figure 12.6). These hedgers would like to go short. Their demands are met by speculators who are willing to go long only if the expected spot price exceeds the current forward price. This is because for the speculators to be in the market, their expected return must be positive. By selling at a price below the expected future spot price, hedgers entice the speculator to go long, and this transfers the unwanted risk to the speculator. This creates a normal backwardation because the forward price is less than the expected future spot price. And, the risk premium is reversed. The concepts of normal contango and normal backwardation are, thus, exactly the notions needed to understand risk premium and whether the forward price exceeds or is less than the expected futures spot price of the commodity. Our cost-of-carry models cannot answer these more subtle questions. The trade-off is that the cost-ofcarry model gives a price, but the economic approach doesn’t.

12.6

Market Imperfections

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How does the cost-of-carry model get modified if we start considering market “imperfections” such as transaction costs, different borrowing and lending rates, and restrictions on short selling? Here the cash-and-carry arbitrage and the reverse cashand-carry are no longer symmetric. Consequently, we get price ranges within which the forward price lies. These bounds are likely close because it’s the trader with the lowest transaction costs (and not retail traders like us) who take advantage of arbitrage opportunities and determine the price bounds. Example 12.5 introduces brokerage commissions into our basic cost-of-carry model.

EXAMPLE 12.5: Forward Price Bounds in the Presence of Market Imperfections ■

Consider the data from Example 11.1. YBM’s stock price S is $100 today (January 1, time 0), and the dollar return (1 + R) is $1.06. Consider a newly written forward on YBM that matures in one year (time T).

■

Assume that when trading YBM, a brokerage commission is charged that equals TC = 1 percent of the stock price. For simplicity, assume that there are no transaction costs on the maturity date. Now we need two tables, Tables 12.5a and 12.5b, for an arbitrage portfolio.

■

Table 12.5a shows, to prevent arbitrage, −101 + 0.9434F ≤ 0 or F ≤ 107.06 This gives an upper bound for the forward price.

(12.15a)

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■

Table 12.5b shows the no-arbitrage portfolio for a reverse cash-and-carry. Here −0.9434F + 99 ≤ 0 or 104.9396 ≤ F

(12.15b)

This gives a lower bound for the forward price. ■

Combining expressions (12.15a) and (12.15b), 104.9396 ≤ F ≤ 107.0596

(12.15c)

The forward price lies between $104.9396 and $107.06. ■

Replacing numbers with symbols gives the formula S (1 + R) (1 − TC) ≤ F ≤ S (1 + R) (1 + TC)

(12.16a)

TABLE 12.5A: Arbitrage Table for Cash-and-Carry Portfolio

Buy spot (pay 1 percent transaction cost today)

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Sell forward Borrow present value of F Net cash flow

Today (Time 0) Cash Flow

Maturity Date (Time 1) Cash Flow

–100 × (1 + 0.01)

S(T)

0

–[S(T) – F]

0.9434F

–F

–101 + 0.9434F

0

TABLE 12.5B: Arbitrage Table for Reverse Cash-and-Carry Portfolio

Sell spot (deduct transaction costs from short sale proceeds) Buy forward Lend present value of F Net cash flow

Today (Time 0) Cash Flow

Delivery Date (Time 1) Cash Flow

100 × (1 – 0.01)

–S(T)

0

S(T) – F

–0.9434F

F

–0.9434F + 99

0

MARKET IMPERFECTIONS

Besides brokerage commissions, there may be a bid/ask spread.4 As transaction costs increase, the price bounds become farther apart. However, different traders may have different transactions costs. This opens up quasi-arbitrage (near-arbitrage) opportunities for the lowest cost trader. One can introduce more market frictions in a similar fashion. For example, with unequal borrowing and lending rates, the relation becomes S (1 + RLEND ) ≤ F ≤ S (1 + RBORROW )

(12.16b)

where 1 + RLEND is the dollar return from lending and 1 + RBORROW is that for borrowing. Several of these relations may be combined.

RESULT 12.5 The Cost-of-Carry Model in the Presence of Brokerage Commissions and Unequal Borrowing and Lending Rates When stock trades are charged a brokerage commission of TC on today’s trade, and the borrowing rates are higher than the lending rates, then the forward price F satisfies:

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S (1 + RLEND ) (1 − TC) ≤ F ≤ S (1 + RBORROW ) (1 + TC)

(12.16a)

where S is today’s stock price, F is today’s forward price for time T, and 1 + RLEND and 1 + RBORROW are the dollar returns for lending and borrowing, respectively.

Like playing with children’s blocks, you can combine Results 12.4 and 12.5 to build even more general models. To apply the cost-of-carry model in practice requires a good understanding of the market and matching the market’s structure to the model’s assumptions. For financial commodities like stock indexes or Treasury securities, convenience yields and storage costs are not relevant, so the simpler models apply. In contrast, for agricultural commodities like corn or wheat, convenience yields and storage costs are extremely important and need to be included. Given the previous discussion, all that is left for a correct application of these models is experience and market savvy. Both of these will come in good time, and the old saying is especially relevant in this regard: “practice makes perfect.”

4

And when you sell short, the brokerage firms may not allow you the full proceeds—you may only earn interest on part of the funds kept with the broker. When there are restrictions on short selling, you may have an inequality like S(1 + q × Interest) ≤ F ≤ S(1 + R), where q is the fraction of usable funds derived from the short sale. Notice that this arbitrage only works in one direction because long and short positions have asymmetric payoffs. This relationship is less important because rules of short selling tend to vary from country to country.

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12.7

Summary

1. We extend the basic cost-of-carry model to incorporate market frictions (relax assumption A1 of Chapter 11) and dividends (ease A4) into the framework. 2. Assuming that interest is the only relevant carrying charge, we get three forward pricing models for assets that generate income:

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a. The simplest dividend model assumes that the underlying stock pays fixed dollar dividends on known dates. This gives Result 12.1A, which states that the stock price net of the present value of dividend is the discounted forward price or S – B1 div = BF. This is useful for finding forward prices for contracts written on dividend-paying stocks and coupon-bearing Treasury bonds that mature after the delivery date. b. When the underlying stock pays dividends at a continuous rate 𝛿, we get Result 12.1B: the forward price is the compounded value of the stock price, where compounding is done at a net rate equal to the difference between the interest rate and the dividend yield or F = Se(r − 𝛿)T . This is useful for finding forward prices when the underlying is an index portfolio. c. A modification of the previous model gives covered interest rate parity, which expresses the forward exchange rate as the spot exchange rate compounded at a net rate equal to the difference between the domestic and the foreign interest rates. Result 12.2 states that FA = SA (BE /B) = SA e(r − rE )T . 3. For commodities mainly used for consumption or production, storage costs can be important. However, holders of the physical commodity may also get a convenience yield. We include these costs and benefits to build a generalized costof-carry model (Result 12.4): the forward price is the compounded value of the spot price, where compounding is done at a net rate equal to the costs (interest rate r plus the storage cost rate g) minus the benefits (dividend yield 𝛿 plus the convenience yield y) or F = Se((r + g)−(𝛿 + 𝛾))T . 4. Contango is a market where the forward price is higher than the spot price. This happens when the net cost-of-carry is positive in the generalized cost-of-carry model. The market is in backwardation when the reverse happens. 5. Normal contango happens when a majority of the hedgers want to set up a long hedge. Speculators assume the other side of the trade, but they require a risk premium for supplying liquidity: the forward price is greater than the expected future spot price. When the net hedgers are short, the risk premium is reversed, and a normal backwardation market is created where the forward price is less than the expected future spot price. 6. Buys and sells are no longer symmetric when market imperfections like brokerage costs, differential borrowing and lending rates, and short selling restrictions are introduced. Well-defined prices give way to ranges within which forward prices lie.

QUESTIONS AND PROBLEMS

12.8

Cases

Diva Shoes Inc. (Darden School of Business Case UV0265-PDF-ENG, Harvard

Business Publishing). The case studies a US-based manufacturer’s currency risk exposure and considers whether hedging via a forward contract or a currency option is advisable. Leveraged Buyout (LBO) of BCE: Hedging Security Risk (Richard Ivey School

of Business Foundation Case 908N23-PDF-ENG, Harvard Business Publishing). The case considers an equity partnership’s currency risk exposure and evaluates various derivative instruments for hedging those risks. Risk Management at Apache (Harvard Business School Case 201113-PDF-ENG).

The case evaluates a company’s hedging strategy and the derivatives used for this purpose.

12.9

Questions and Problems

12.1. a. Explain the cost-of-carry model with dollar dividends in your own words. b. Justify why the spot price considered in the model is net of the present value

of all future dividends paid over the life of the contract. c. Why don’t we adjust for dividends that are paid after the forward’s maturity

date?

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12.2. Boring Unreliable Gadget Inc.’s stock price S is $50 today. It pays a dollar

dividend after two months. If the continuously compounded interest rate is 4 percent per year, what is the forward price of a six-month forward contract on BUG? 12.3. Boring Unreliable Gadget Inc.’s stock price S is $50 today. It pays dividends

of $1 after two months and $1.05 after five months. If the continuously compounded interest rate is 4 percent per year, what is the forward price of a six-month forward contract on BUG? 12.4. Boring Unreliable Gadget Inc.’s stock price S is $50 today. The company,

however, reliably pays quarterly dividends to shareholders. For example, BUG paid $0.95 dividend one month back; it will pay $1 dividend two months from today, $1.05 after five months, $1.10 after eight months, and $1.15 after eleven months. If the continuously compounded interest rate is 4 percent per year, what is the forward price of a six-month forward contract on BUG? 12.5. Boring Unreliable Gadget Inc.’s stock price S is $50 today. It pays dividends

of $1 after two months and $1.05 after five months. The continuously compounded interest rate is 4 percent per year. If the six-month forward price is $51, demonstrate how you can make arbitrage profits or explain why you cannot.

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12.6. Boring Unreliable Gadget Inc.’s stock price S is $50 today. It pays dividends

of $1 after two months and $1.05 after five months. The continuously compounded interest rate is 4 percent per year. Transactions costs are $0.10 per stock traded, a $0.25 one-time fee for trading forward contracts, and no charges for trading bonds. If the six-month forward price is $51, demonstrate how to make arbitrage profits or explain why you cannot. 12.7. Explain how the cost-of-carry model with dollar dividends differs from the

cost-of-carry model with dividend yields. 12.8. What is a convenience yield for a commodity and why is it important to

include in the cost-of-carry model? Give an example of a commodity that has a convenience yield. 12.9. a. What is a stock index? What happens to stock indexes when dividends are

paid? b. What is a synthetic index? Why is it useful? 12.10. Consider SINDY Index obtained by averaging stock prices and a synthetic

index SINDY spot that replicates its performance. (1) SINDY’s current level I is 10,000, and the synthetic index’s price S is $10,000. (2) A newly written forward contract on the index matures after T = 0.5 years. (3) The continuously compounded interest rate r is 5 percent per year. (4) Stocks constituting SINDY spot paid $190 of dividends last year and are expected to pay the same this year. a. Compute the dividend yield 𝛿 on SINDY spot.

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b. Compute the forward price F. 12.11. Consider the data given in problem 12.11 for futures contracts on SINDY

Index. If the forward price is F = $10,231, show how you can make arbitrage profits or explain why you cannot. 12.12. Consider the data given in problem 12.11 for futures contracts on SINDY

Index. Suppose you have to pay transaction costs: (1) When you go long or sell short a portfolio of stocks, transaction costs (brokerage fees plus the price impact of the trade) equal 10 basis points of the synthetic index’s price. (2) There is a one-time fee of $15 for trading forward contracts but no charges for trading bonds. If the forward price is F = $10,231, show how you can make arbitrage profits or explain why you cannot. 12.13. Today’s spot exchange rate SA is $1.30 per euro in American terms. The

continuously compounded annual risk-free interest rates are r = 4 percent in the United States (domestic) and rE = 3 percent in the Eurozone. What is the four-month forward rate in American terms if the cost-of-carry model holds?

QUESTIONS AND PROBLEMS

12.14. The current dollar/Swiss franc spot exchange rate is 0.5685. If you invest one

dollar for ninety days in the US domestic riskless asset, you earn $1.0101, and if you invest one franc for ninety days in the Swiss riskless asset, you earn 1.0113 francs (assume continuous compounding). A broker offers you a ninety-day forward contract to buy or sell 1 million francs at the exchange rate of 0.55 dollars/franc. Are there arbitrage profits to be made here? If so, compute them. 12.15. Alloyum costs $0.10 per month to store (which is paid upfront) but gives a

convenience yield of $0.12 per month (which is received on the maturity date). If its spot price S is $200 per ounce and the continuously compounded interest rate r is 5 percent per year, what is the five-month forward price for Alloyum? 12.16. In the previous example, suppose a trader quotes a five-month forward price

of $206 per ounce. Demonstrate how you can make arbitrage profits from these prices; if you cannot, explain your answer. 12.17. COMIND Index is computed by averaging commodity prices. Compute

the five-month forward price for this index if the spot price is 1,000 and the continuously compounded annual rates for various costs and benefits are 5 percent for the interest rate, 3 percent for the dividend yield, 4 percent for the storage cost, and 3 percent for the convenience yield. 12.18. Find the price bounds for the five-month forward price for Boring Unreliable

Gadgets when

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a. BUG’s stock price S is $50 today b. a trader can borrow money at 5 percent and lend money at 4 percent,

where the interest rates are annual simple interest rates c. a brokerage commission of 0.5 percent of the stock price is charged today

but your broker waives transaction costs on the maturity date 12.19. Forward prices for April and June forward contracts for platinum are $400 per

ounce and $410 per ounce, respectively. The interest rate is 1 percent for the April–June period. There is a 1 percent transaction cost whenever you trade (buy or short sell) the spot in June, but the broker is waiving any transaction costs for trading forward contracts and brokerage costs for trading the spot in April. Can you generate arbitrage profits? Explain your answer. (Assume that the spot price of platinum in June lies between $400 and $500 per ounce.)

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13 Futures Hedging 13.1 Introduction

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13.2 To Hedge or Not to Hedge

13.5 Futures versus Forward Hedging

Should the Firm or the Individual Hedge?

13.6 Spreadsheet Applications: Computing h

The Costs and Benefits of Corporate Hedging

13.7 Summary

EXTENSION 13.1 Airlines and Fuel Price Risk EXTENSION 13.2 A Hedged Firm Capturing a Tax Loss

13.3 Hedging with Futures

13.8 Appendix Deriving the Minimum-Variance Hedge Ratio (h) Computing the Minimum-Variance Hedge Ratio (h)

Perfect and Imperfect Hedges

Statistical Approach

Basis Risk

Econometric Approach: A Linear Regression Model

Guidelines for Futures Hedging

13.4 Risk-Minimization Hedging The Mean-Variance Approach Limitations of Risk-Minimization Hedging

13.9 Cases 13.10 Questions and Problems

TO HEDGE OR NOT TO HEDGE

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13.1

Introduction

The novelist Fyodor Dostoyevsky perceptively observed in The Gambler: “as for profits and winnings, people, not only at roulette, but everywhere, do nothing but try to gain or squeeze something out of one another.” This aptly defines a zero-sum game of which, excluding transaction costs, futures trading is an excellent example. There is a pool of talented futures traders who earn a living from this business thereby depressing the chances of winning for ordinary traders who lose money on average. Then, why trade futures? Cynics give a facile explanation: people are “into commodities” because it’s as thrilling as going to the casinos. True to circus-owner P. T. Barnum’s immortal quote that “there’s a sucker born every minute,” these gamblers gladly lose for the joy of the ride. Perhaps others believe that they really can win the futures game. We know no surefire strategy. Even if you stumble on such a strategy, repeated use will destroy its efficacy. So, why trade futures? In reality, most futures contracts are traded to hedge risks that preexist in some line of business, and hedgers are willing to give up expected returns as payment for this “insurance.” We discussed this issue in the context of normal backwardation in Chapter 12. For example, Figure 13.1 shows a man selling goods on some tourist spot, perhaps a beach. If he chooses his wares wisely, say, by carrying an assortment of umbrellas and sunglasses, then he has hedged well. Come rain or come shine, he has something to sell. If this is not possible, he could use futures to manage these risks. For example, he could sell only umbrellas and buy futures on a sunglass company’s stock, instead of selling sunglasses. Who hedges? A majority of the Fortune 500 companies and a growing number of smaller firms hedge risk. Farmers often hedge the selling price of their produce. Many producers hedge input as well as output price risk. This chapter explores the reasons for hedging, discusses its cost and benefits, and introduces futures hedging strategies. We discuss perfect and cross hedges, long and short hedges, and risk-minimization hedging using standard statistical techniques.

13.2

To Hedge or Not to Hedge

Unfortunately, there is no general answer to what may be phrased as (with apologies to the immortal William Shakespeare) to hedge or not to hedge—that is the question. We can only discuss hedging’s benefits and costs.

Should the Firm or the Individual Hedge? The classic Modigliani and Miller (M&M) papers argued that debt policies do not affect firm value. This irrelevance follows because a shareholder can replicate such policies herself by trading stocks in otherwise identical firms. Therefore the noarbitrage principle ensures that these companies, differing only in their debt policies, must have equal value. The same argument can be applied to a firm hedging its balance sheet risks giving intellectual support to the irrelevance of hedging. However, the M&M results rely on several key assumptions, including no market frictions

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FIGURE 13.1: Wisely Hedged: Come Rain, Come Shine, He Has Something to Sell*

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Umbrellas $10

Umbrellas $10 Sunglasses $15 Inspired by a 1980s newspaper advertisement.

(including bankruptcy costs), no taxes, and no information asymmetry. In the real world, these assumptions fail to hold. Consequently, if reducing the firm’s risk is in the shareholders’ best interest, then a firm can usually do a better job of hedging than individual investors. Indeed, a company (1) is likely to have lower transaction costs; (2) can trade larger contracts than a shareholder; (3) can dedicate competent personnel to hedging; (4) can hedge by issuing a whole range of securities, which individuals cannot; (5) may have private information about the company’s risks; and (6) can hedge for strategic reasons that lie beyond an ordinary investor’s knowledge. In all these cases, a company creates more value than an individual investor hedging on her own. But hedging is costly to implement, both in developing and retaining the relevant expertise within the firm, and in executing the transactions.

TO HEDGE OR NOT TO HEDGE

The Costs and Benefits of Corporate Hedging A hedged business has fewer risks to worry about, but exactly why is this an advantage? Aren’t businesses supposed to take risks? We answer these questions next, and although we focus on the use of futures, many of these same benefits can be obtained with other derivatives such as forwards, options, and swaps: 1. Hedging locks in a future price. Recall Chapter 8, in which we discussed the establishment of the Chicago Board of Trade. The CBOT built orderly markets for grain trading to avoid price crashes at harvest times and booms afterward. Hedging with futures helps smooth such demand–supply imbalances. It allows traders to lock in stable prices and plan production and marketing activities with greater certainty.

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2. Hedging permits forward pricing of products. For example, many airlines routinely hedge aviation fuel prices, which is a major input cost (see Extension 13.1 for a discussion). If fuel costs can be fixed (and other costs, such as staff salary, airplane depreciation, airport gate rental charges, and travel agents’ commissions, can be estimated in advance), then the airline can better set a profit margin and determine future seat prices. Customers like knowing travel costs far in advance. 3. Hedging reduces the risk of default and financial distress. Default means a failure to pay a contractual payment, such as debt, when promised. Default triggers unfavorable outcomes such as lawyer fees, potential bankruptcy costs and liquidation fees, losing control over the company’s assets, and increased costs of doing business because suppliers fear that they may not be paid and customers may worry about quality and service. These are called financial distress costs. A company can reduce the likelihood of incurring these costs by hedging some of the risks it faces. 4. Hedging facilitates raising capital. Because of the decreased risk of default, bankers allow hedgers to borrow a larger percentage of a commodity’s value at a lower interest rate than nonhedgers. This same logic applies to firms. 5. Hedging enables value-enhancing investments. Froot et al. (1994) argue that if external sources of funds (like stock and bond issuances) are costlier to corporations than internally generated funds, then hedging can help stabilize internal cash flows and make them available for attractive investment opportunities. 6. Hedging reduces taxes. A hedge may be used to capture the benefits of a tax loss or take advantage of a tax credit. The basic intuition is simple. Because one can use a tax loss or a credit only when the company has positive profits, it may be worthwhile to smooth out profits by hedging rather than letting them fluctuate. See Extension 13.2 for an example that illustrates this notion. However, hedging also incurs costs. When trading with a speculator, a hedger often pays an implicit fee by trading at a price inferior to the expected future payoff. Of course, brokers must also be paid. The presence of these trading costs constitutes a basic argument against futures hedging. Moreover, businesses must allocate valuable personnel to devise hedging strategies, set up adequate checks and

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balances to prevent rogue employees from ruinously speculating with the firm’s money, ensure compliance with the laws of the land, and meet the government’s accounting requirements in this regard.1 Hedging is analogous to purchasing an insurance policy on some commodity’s spot price. It pays off when spot prices move in an adverse direction. However, as with all insurance policies, there is a cost—the premium. If the spot price does not move adversely, one pays for insurance not used. Risk-averse individuals buy insurance despite this cost, and analogously, firms often hedge. To hedge or not to hedge remains an unsettled question that must be resolved on a case-by-case basis. One often hears any number of catchy maxims: “no risk, no gain” alongside “risk not thy whole wad.” Which to choose requires expertise, finesse, and knowledge, which are acquired through study and experience.

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EXTENSION 13.1: Airlines and Fuel Price Risk Fuel costs can hit the airlines hard. Fuel is a major cost of the airline business; it can be between 10 percent (in good times) and more than 35 percent (in bad times) of their average expenses.2 “When the going gets tough, the tough get going” goes a familiar saying. Crude oil prices had a huge run-up in 2008 and peaked at over $145 per barrel during July of that year. As the oil price went up, the airlines adopted a wide range of measures, some prudent and others drastic, to reduce fuel consumption. Most of these approaches aimed at lowering the aircraft’s weight because a lighter plane is cheaper to fly. Airlines started charging for checked-in luggage, which “killed two birds with one stone”: raised money and reduced weight. Other steps included flying more fuel-efficient planes and reducing the time that plane engines stay on. Some started trading commodities. Yet, crude oil prices started falling and jet fuel prices also declined. In November 2008, crude prices closed below $60 per barrel. A report in the New York Times entitled “Airlines find it difficult, and perhaps unwise, to hedge fuel prices” dated November 13, 2008 reported that the German airline Lufthansa’s “hedging for the current year had been cut to 72 percent from 85 percent of its total fuel bill ... Its hedging for 2009 is down to 57 percent, at an average of $91 a barrel.” Stunned by unprecedented oil price volatility, several other airlines were left holding costly hedges. Many commercial airlines as well as companies like FedEx (which uses its extensive fleet to quickly deliver packages to numerous locations around the globe) use derivatives on oil to hedge. The hedged amount varies. Sometimes they hedge little or none of their exposure; at other times they hedge much more (see Table 13.1 for a description of the fuel hedging at some of the world’s largest airlines based on thier annual reports). Southwest Airlines has a history of running a successful fuel “hedging program.” It has maintained its long track record of profitability into the new millennium by extensive hedging even at a time when many other airlines were reeling from losses and operating under bankruptcy protection. Consider the 2016 annual report of Southwest Airlines, which states “Because the Company uses a variety of different derivative instruments at different price points, the Company is subject to the risk that the fuel derivatives it uses will not provide adequate protection against

1 Barings Bank, one of England’s oldest and most prestigious merchant banks, founded in 1762 by Sir Francis Baring, collapsed in 1995 after a rogue employee, Nick Leeson, lost $1.4 billion of company money speculating in futures contracts. Nowadays, most companies have stronger oversight of the firm’s overall risk situation through the office of a chief risk management officer, who often reports directly to the chief executive officer.

TO HEDGE OR NOT TO HEDGE

261

significant increases in fuel prices and could in fact result in hedging losses, and the Company effectively paying higher than market prices for fuel, thus creating additional volatility in the Company’s earnings. … Excluding the impact of hedging, Fuel and oil expense would have decreased by $535 million, or 15.9 percent, compared with 2015, due to lower market jet fuel prices.”

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TABLE 13.1: Fuel “Hedging Program‘ at some of the World’s Largest Airline Companies by Revenue (in US dollars) Airlines

Hedging Policy (as of December 2016 or for the year ending December 31, 2016)

American Airlines Group (USA)

No fuel hedges as of December 31, 2016; fully exposed to fluctuations in fuel prices.

Delta Air Lines (USA)

Manages fuel price risk through a “hedging program” with options, swaps and futures on commodities including crude oil, diesel fuel and jet fuel. Recorded fuel hedge losses of $366 million in 2016 and $741 million in 2015.

United Continental Holdings (USA)

The Company historically hedges a portion of its planned fuel requirements. No outstanding fuel hedges as of December 31, 2016. Although the Company’s current strategy is to not enter into transactions to hedge its fuel consumption, it regularly reviews its strategy based on market conditions and other factors.

Deutsche Lufthansa (Germany)

Fuel costs included a loss due to hedging equal to EUR 905m (One Euro was $1.05 on December 31, 2016). Fuel expenses were about EUR 4.9 billion in 2016. The Lufthansa Group hedges fuel prices with a time horizon of up to 24 months.

Air France-KLM (France)

Based on the forward curve at December 31, 2015, a hedging strategy, approved by the Board of Directors, sets the hedge horizon at two years (a rolling 24 months) and the target hedge ratio at 60%. The hedging uses simple futures or option-based instruments.

International Airlines (United Kingdom)

Fuel price risk is partially hedged through the purchase of oil derivatives in forward markets. The current strategy, as approved by the IAG Management Committee, is to hedge a proportion of the anticipated fuel consumption for the next eight quarters. The company primarily trades forward crude, gas oil and jet kerosene derivative contracts for hedging purposes.

Southwest Airlines (USA)

Discussed in text.

Southwest’s “hedging” experience demonstrates the difficulties in distinguishing hedging from speculation and illustrates that even well-considered actions do not always deliver favorable outcomes. 2

As it is extremely difficult to predict oil prices, it is equally difficult to devise an airline’s fuel hedging strategy, which many airlines refer to as a “hedging program.”

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EXTENSION 13.2: A Hedged Firm Capturing a Tax Loss Hedging with derivatives expands a company’s choices and may allow it to take advantage of tax situations. The next example shows how futures contracts can be used to stabilize earnings so that a company can utilize past losses to reduce current taxes.

EXT. 13.2 EX. 1: Hedging to Capture a Tax Loss ■

Goldmines Inc. (fictitious name) mines, refines, and sells gold in the world market. The company’s profits move in tandem with gold price movements. Assume that the pretax profit is $100 million when gold prices go up (which happens with a 50 percent chance) and 0 if gold prices go down (which also happens with a 50 percent chance). Alternatively, Goldmines can hedge with gold futures and have a known profit of $48 million.

■

Ext. 13.2 Fig. 1 shows these payoffs. We label them as event 1 (profit $100) and event 2 (profit $0). These events are shown in a binomial tree, where the payoff $100 is placed on the upper branch of the tree, while $0 is placed on the lower branch.

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Expected Profit ■

The expected profit = [(Probability of high gold price) × (Payoff $100)] + [(Probability of low gold price) × (Payoff $0)] = 0.50 × $100 + 0.50 × 0 = $50 for the unhedged company.

■

We can show the superiority of the hedging strategy by computing the expected after-tax profits for the unhedged and hedged firm under different scenarios.

After-Tax Expected Profits (Simple Case) ■

Assume that the tax rate is 30 percent. When the gold price is high, $100 million pretax profit gives an after-tax profit of 100 × (1 – tax rate) = $70 million. When the gold price is low, the pretax profit is 0, and so is the after-tax profit. After-tax profits are shown in Ext. 13.2 Fig. 1. Expected after-tax profit for unhedged firm = 0.50 × 70 + 0.50 × 0 = $35 million

■

The hedged firm makes a profit of $48 million irrespective of the gold price: After − tax profit for hedged firm = 48 × 0.7 = $33.6 million

TO HEDGE OR NOT TO HEDGE

■

263

The company can choose either. The actual choice depends on the risk preferences of the company’s management and shareholders.

After-Tax Expected Profits (When $25 Million Tax-Deductible Loss Is Carried Forward) ■

Now suppose that the company has accumulated losses totaling $25 million. It can deduct this loss from this year’s profit and thus lower the tax burden. Such strategies of tax reduction are known as tax shields.

■

Assume that if unutilized, this one-shot opportunity disappears. We will show that the hedged firm can always utilize the losses to lower its taxes, but the unhedged firm can only do this half the time.

■

For the unhedged firm, total tax when the gold price is high is (Pretax profit − Loss deducted) × (Tax rate) = (100 − 25) × 0.3 = $22.5 million

EXT. 13.2 FIG. 1: Profits of a Hedged and an Unhedged Firm Pre-Tax Profit 0.5

$100.00 Hedged Firm

Unhedged Firm

$48

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0.5

0 Expected profit of the unhedged firm = 0.5 × 100 + 0.5 × 0 = $50 million Expected After-Tax Profit (with No Loss Carried Forward) 0.5

$70.00

Unhedged Firm

Hedged Firm

$33.60

0.5

0 Expected profit = $35 million Expected After-Tax Profit (with $25 Million Tax Deductible Loss) 0.5

$77.50

Unhedged Firm

Hedged Firm 0.5

0 Expected profit = $38.75 million

$41.10

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The total tax when the gold price is low is 0. Consequently, the after-tax profit is (100 – 22.5) = $77.5 million when the gold price is high and zero when it is low (see Ext. 13.2 Fig. 1): Expected after-tax profit for the unhedged firm = 0.50 × 77.5 + 0.50 × 0 = $38.75 million ■

By contrast, the total tax for the hedged firm is (48 – 25) × 0.3 = $6.90: After-tax profit for the hedged firm = (48 – 6.90) = $41.10 million

■

Clearly hedging is a superior strategy: not only does the hedged firm generate greater after-tax profit but it also removes swings and stabilizes this amount.

13.3

Hedging with Futures

Chapter 4 gave us an inkling of how a company can use forward contracts for hedging input and output price risks. Now that you are more familiar with the workings of a futures contract (which are easier to use than forward contracts), let us explore hedging with futures.

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Perfect and Imperfect Hedges Ours is an imperfect world, in which it is hard to find perfect hedges. A perfect hedge completely eliminates spot price risk for some commodity. This happens when (1) the futures is written on the commodity being hedged, (2) the contract matures when you are planning to lift the hedge, and (3) the contract size and the other characteristics of a futures unerringly fit the hedger’s need. Imperfect or cross hedges occur when these three conditions are not satisfied. For example, a bank may reduce the price risk of its loan portfolio, which is vulnerable to interest rate changes, by trading Treasury bond futures contracts. This is a cross hedge because the futures is written on US Treasury bonds, whereas the loans being hedged consist of, say, house mortgages, car loans, and certificates of deposit. Even when futures on the same commodity are available, the issue of a timing mismatch may occur. Consequently, basis risk emerges as a paramount concept in analyzing, setting, and managing a futures hedge.

Basis Risk Basis risk is a focal point in understanding futures hedges. The basis is defined as the difference between the spot and the futures price. It is written as Basis = Cash price – Futures price. Example 13.1 shows how crucial basis risk is when hedging a

HEDGING WITH FUTURES

commodity’s spot price risk. We illustrate this in the context of two companies using buying and selling hedges to offset input and output risks, respectively.

EXAMPLE 13.1: A Gold Futures Basis Risk ■

■

Today is January 1, which is time 0. Consider the gold futures contract trading on the COMEX Division of the CME Group described in Chapter 8. The prices are reported on a per ounce. Figure 13.2 gives the timeline and the various prices. For simplicity, we assume that no interest is earned on margin account balances. On today’s date, the spot price of gold S = $990, and the June contract’s futures price F(0) = $1,000. The basis is b (0) = Spot price S (0) − Futures price F (0) = 990 − 1,000 = −$10

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FIGURE 13.2: Timeline and Prices for Gold Futures Hedging Example

■

Start date (January 1) Time 0

Closing date (May 15) Time T

Delivery period (June)

Spot price, S(0) = $990 Futures price, F (0) = $1,000 Basis, b(0) = S(0) – F (0) = –$10

S(T) = $950 F (T) = $952 b(T) = S(T) – f (T) = –$2

The delivery period for the contract is in June. For the subsequent discussion, suppose that on May 15, time T, the spot price S(T) is $950 and the June futures price F(T) is $952. The new basis is b (T) = S (T) − F (T) = 950 − 952 = −$2

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Goldmines Inc. Sets Up a Short Hedge ■

■

Suppose that the mining company Goldmines Inc. goes short one June gold futures contract on January 1 to hedge its output price risk. The company sells gold and lifts the hedge by closing out the futures position on May 15. Goldmines sells gold for $950, but it makes – [F(T) – F(0)] = – (952 – 1,000) = $48 on the futures position. So, the effective selling price on May 15 is (see Equation 9.1b of Chapter 9) Spot cash flow + Futures cash flow = 950 + 48 = $998

■

One can rewrite this as Spot − (Change in futures price) = S (T) − [F (T) − F (0)] = [S (T) − F (T)] + F (0) = New basis + Old futures price

(13.1)

No matter when the company closes its position, the old futures price is fixed, and only the new basis affects profits.

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Jewelrygold Inc. Sets Up a Long Hedge ■

Now look at the same problem from the perspective of a jewelry maker, Jewelrygold Inc. As discussed in chapter 4, the company can reduce input price risk by setting up a long hedge (a buying hedge) and going long gold futures. When it is time to buy gold in the spot market, the company sells these futures and removes the hedge.

■

On January 1, Jewelrygold goes long a June gold futures contract to hedge input price risk. On May 15, the company buys gold for $950 and simultaneously sells futures for $952 to end the hedge. Although Jewelrygold pays less for gold, it has to surrender nearly all of its gains through the hedge. The cash flow from the futures position is [F(T) – F(0)] = 952 – 1,000 = –$48.

■

Jewelrygold’s effective price on May 15 is $998. By looking at the cash flows, the buying price is = −Spot + (Change in futures price) = −S (T) − [F (T) − F (0)] = − [S (T) − F (T) + F (0)] = − (New basis + Old futures price)

(13.2)

This is the same value as in the previous example, except for the minus sign.

This example shows that a futures hedged spot commodity’s portfolio value can always be viewed as the sum of the old futures price and the new basis. Consequently, hedgers are interested in how the basis evolves randomly through time. This randomness gives rise to basis risk in hedging, which is often measured by computing the basis’s variance (or standard deviation). Basis risk is, thus, fundamental

HEDGING WITH FUTURES

to futures hedging, and hedgers often talk about the widening or narrowing of the basis in passionate terms. Many of them have extensive charts depicting the historic behavior of the basis, looking for a crystal ball to foretell the future.

Guidelines for Futures Hedging Real life is far more complex than textbook examples. Exchanges offer only a handful of futures, whereas there are thousands of commodities. The specter of an imperfect hedge visits us again. When setting up a hedge, it’s natural to ask, which contract and for what maturity? Example 13.2 discusses the issues involved in answering this question.

EXAMPLE 13.2: Hedging by Selecting a Futures from Several Competing Contracts Suppose that a new alloy Alloyum (a fictitious name) is developed that can replace precious metals in industrial and ornamental use. Let Alloyum trade in the spot market. Alloyum futures may or may not trade, and we discuss both possibilities. If you produce thirty thousand ounces of Alloyum, how would you hedge output price risk?

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Alloyum Futures Trade, and You Know When to Lift the Hedge This case happens when the Alloyum futures contract’s maturity date exactly matches the delivery date of the spot commodity commitment. Then, sell [(Spot position)/(Alloyum futures contract size)] = 30,000/50 = 600 Alloyum futures, assuming a contract size of fifty ounces. Basis risk eventually disappears because the spot converges to the futures price at maturity. This can be shown by setting S(T) = F(T) in Equation 13.1. With zero basis risk, you fix the selling price at maturity, which is none other than today’s futures price. Congratulations, you have set up a perfect hedge. (This perfect hedge ignores the interest rate risk from marking-to-market. Such reinvestment risks will be considered later in the chapter.)

Alloyum Futures Trade, and You Do Not Know When to Lift the Hedge In this case, one way to proceed is to compute the variance of the basis for different maturity futures contracts on Alloyum and select the one that scores the lowest. Lower basis risk means that the futures price deviates less from the spot price. You are likely to find that the futures contract maturing in the same month as the spot sale is the best candidate. In this case, hedge with the smallest basis risk futures contract. The number of contracts to be shorted may be found by the risk-minimization hedging strategy discussed later in this chapter.

Alloyum Futures Do Not Trade, and You Know When to Lift the Hedge In this case, you may first like to select a similar commodity on which a futures contract is written and decide on the maturity month. Collect price data for Alloyum and for contender commodity futures contracts maturing in the same month as the spot sale, and compute correlations of price changes. Select the commodity futures that has the highest correlation to Alloyum in the overall period.

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Alloyum Futures Do Not Trade, and You Do Not Know When to Lift the Hedge This is the worst case for hedging. First, select a commodity futures contract as in the previous case. Then, select a maturity month, and find the futures contract with the largest correlation with Alloyum spot price changes and that minimizes the variance of the basis. There is, however, a famous saying in economics that “there are no solutions, only trade-offs” (Sowell 1995, p 113). The other futures contracts may have lower transaction costs. Liquidity is another concern because it often increases as the contract approaches maturity, but dries up in the delivery month.

13.4

Risk-Minimization Hedging

Suppose that you want to hedge a long spot commodity position by shorting futures contracts. Using a statistical or econometric model, assuming that past price patterns repeat themselves enables you to find the optimal number of contracts to minimize the variance of the hedging error. This section discusses this statistical approach to hedging.

The Mean-Variance Approach The mean-variance approach to risk-minimization hedging (or optimal hedging) determines the optimal number of contracts needed to minimize the variance of the portfolio’s price changes. The intuition of this approach can be understood by first considering a perfect hedge: Copyright © 2018. World Scientific Publishing Company Pte. Limited. All rights reserved.

■

A perfect hedge. Suppose you are holding a long position in the spot commodity. You need to short futures contracts to hedge the spot commodity’s price risk. In the pristine world of a perfect hedge, any change in the spot position will be exactly offset by an equal and opposite change in the futures position. Consequently, Change in portfolio value = Change in spot position – Change in futures position = 0

(13.3)

The minus sign before the change in the futures position is because we are short the futures contract and the spot and futures prices move in the same direction. With minor modifications, we can recast the problem to focus on how changes in the spot and futures prices per unit are related. Using data from Example 13.2, we can hedge a long position of thirty thousand ounces of Alloyum production by selling [(Cash position)/(Alloyum futures contract size)] = 30,000/50 = 600 Alloyum futures. We can write this as follows: Change in portfolio value = (30, 000 × Change in spot price per ounce) − (600 × 50 × Change in futures price per ounce) =0 (13.4)

RISK-MINIMIZATION HEDGING

There is no change in the portfolio value because the spot and the futures price changes exactly match. ■

An imperfect hedge. In reality, the hedge will be imperfect because the spot and the futures price changes do not exactly match. Here you can find the risk-minimizing number of futures contracts. Consider hedging our spot exposure of n ounces by selling q futures contracts of size f ounces per contract. Then we can rewrite the left side of Equation 13.4 as Change in portfolio value = (n × Change in spot price per ounce) − (q × f × Change in futures prices per ounce) = n [S (t) − S] − qf [F (t) − F] (13.5) = nΔS − qfΔF where S(t) and S are the spot prices per ounce on some future date (time t) and today (time 0), F(t) and F are futures prices per unit on those same respective dates, and Δ compactly denotes the price change. Note that this change does not equal zero because it is an imperfect hedge. To find the number of contracts to sell to risk-minimize hedge the spot portfolio, compute the variance of the portfolio and select n to minimize this variance. You can do this by taking the partial derivative of the portfolio variance with respect to q, setting it equal to zero, and solving for q (see the appendix to this chapter). This leads to the following result (see Result 13.1).

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RESULT 13.1 The Risk-Minimizing Number of Futures Contracts for Hedging a Spot Commodity Position To minimize the risk of a long portfolio of a spot commodity, sell short q contracts where q = (n/f) h (13.6) and n is the size of the spot position, f is the number of units of the underlying commodity in one futures contract, and h is the optimal hedge ratio (or minimum-variance hedge ratio or risk-minimized hedge ratio). The optimal hedge ratio is given by h=

covS,F sdS = corr varF sdF S,F

(13.7)

where covS,F is the covariance between changes in the spot price (ΔS) and the futures price (ΔF), varF is the variance of the change in the futures price, sdS is the standard deviation (or volatility) of the change in the spot price (standard deviation is the positive square root of the variance), sdF is the

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standard deviation of change in the futures price, and corrS,F is the correlation coefficient between ΔS and ΔF.c c

The second part of the formula follows from the result: “covariance equals the product of the standard deviations and the correlation coefficient” or covS,F = sdS × sdF × corrS,F .

Considering various cases reveals the intuition behind this result. Suppose that the price changes for the spot commodity and the futures contract are perfectly correlated (i.e., corrS,F equals 1) and their standard deviations match (i.e., sdS = sdF ). Then the optimal hedge ratio h equals 1. This is the holy grail of a perfect hedge, which is nearly impossible to attain in practice but is useful as an illustration. Next assume that there is a high correlation between the price changes (corrS,F is close to one) and the spot price change has a lower volatility than the futures price change (i.e., sdS < sdF ). Then a unit of spot is hedged with less than a unit of futures. Conversely, if the spot price change fluctuates more than the futures price change (i.e., sdS > sdF ), then one unit of spot needs more than one unit of the futures to hedge. The optimal hedge ratio gives us an easy formula for setting up a futures hedge. Example 13.3 shows how to implement this by utilizing spot and futures price data, which can be easily collected from business newspapers or Internet data sources.

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EXAMPLE 13.3: Setting Up a Risk-Minimization Hedge with Price Data ■

Suppose that you are planning to sell thirty thousand ounces of Alloyum at some future date. To start, you collect Alloyum spot price and platinum futures price data for sixteen consecutive trading days (see Table 13.2).

■

Begin by computing the price differences for each of these two series of price observations. Next use a spreadsheet to compute the parameter estimates needed for the optimal hedge ratio. (The appendix to this chapter discusses related issues and demonstrates how to find h with the help of a simple calculator; Section 13.6 shows how to do these calculations using the spreadsheet program Microsoft Excel.) We get

■

the standard deviation of changes in the futures price per unit, sdF = 14.3368 the standard deviation of changes in the spot price sdS = 15.5002 the correlation coefficient between the spot and futures price changes, corrS,F = 0.7413 Use these estimates in the second part of Equation 13.7 of Result 13.1 to compute the minimumvariance hedge ratio: sdS 15.5002 h= corrS,F = 0.7413 = 0.80154 14.3368 sdF

RISK-MINIMIZATION HEDGING

To hedge n = 30,000 ounces of Alloyum with a platinum futures contract (contract size f = 50), you need to sell

■

q = (n/f) h = (30, 000/50) 0.8015 = 480.9 or 481 contracts (with integer rounding) 4 Or, you can use the first part of Result 13.1 to determine the optimal hedge ratio, h = cov /var = 18.3048/22.8381 = S,F F 0.8015.

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TABLE 13.2: Alloyum Spot and Platinum Futures Price Data Day t

Alloyum Spot S(t)

Platinum Futures F(t)

0

1,215

1,233

1

1,209

1,236

2

1,239

1,245

3

1,245

1,254

4

1,254

1,272

5

1,227

1,254

6

1,230

1,248

7

1,224

1,242

8

1,239

1,260

9

1,251

1,263

10

1,227

1,254

11

1,215

1,227

12

1,209

1,221

13

1,203

1,230

14

1,185

1,203

15

1,194

1,209

Alternatively, you can compute the hedge ratio by suitably framing the problem and estimating a linear regression (see the appendix to this chapter).

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Limitations of Risk-Minimization Hedging Comedian Rodney Dangerfield often chuckled, “I don’t get no respect.” Doesn’t this comment apply to risk-minimization hedging? It is purely computational, and except for some ad hoc adjustments, it is a single-shot exercise with no prescription for modifying the hedge over time. It works well if a closely related commodity can be found and you know when to lift the hedge. It works less well otherwise and if the hedge needs to be rolled over or rebalanced. The idea of rebalancing is related to the concept of (exact) dynamic hedging, which involves regular adjustments to a perfect hedge over time. Dynamic hedging requires sophisticated analytical tools, and it applies in a complete market. It is impossible to implement in an incomplete market (see Duffie [1989] and Jarrow and Turnbull [2000] for examples and issues related to dynamic hedging). In complete markets (remember Chapter 8), all kinds of securities trade that generate future payoffs contingent on all possible future events. In reality, the markets are incomplete, and it may be difficult to develop good futures pricing models to which dynamic hedging applies. Consequently, risk-minimization hedging, despite all its limitations, starts looking respectable again. In incomplete markets, it continues to attract serious research interest.

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13.5

Futures versus Forward Hedging

It’s common practice in many circles to treat forwards and futures as equivalent and interchangeable contracts. However, this is incorrect for many reasons. Although a forward is easy to price, ordinary investors rarely trade in the “organized” forward market. Big banks, large corporations, and other institutions with excellent credit ratings dominate this market, and to guarantee execution of the contracts, these financial institutions usually need to post collateral. These trading restrictions are imposed to reduce “counterparty credit risk”—the nonexecution of the contract terms. Counterparty credit risk is a big concern in the trading of over-the-counter derivatives, as recently evidenced by the related regulatory reforms following the 2007 credit crisis in the Dodd–Frank Wall Street Reform and Consumer Protection Act. Second, given that forward contracts are bilateral negotiated agreements, forward markets are illiquid and subject to significant liquidity risk. For example, if a counterparty closes a forward contract early, significant closing costs are incurred. By contrast, a futures (1) is an exchange-traded, standardized contract; (2) has margins and daily settlement, which makes it safer than a forward due to the absence of credit risk; (3) allows small traders, weaker credits, and complete strangers to participate; and (4) usually trades in a liquid market, where traders can enter or unwind their positions with ease. But there is another important difference between futures and forward contracts due to the marking-to-market of a futures contract. Marking-to-market a futures contract introduces risks involved with reinvesting the cash flows before the contract matures. These same reinvestment risks are not faced by a forward contract. Tracking

SPREADSHEET APPLICATIONS: COMPUTING h

the cash flows to these contracts, the futures trader, unlike the forward trader, earns random and uncertain interest on margin balances. As discussed in Chapter 9, the interest earned on a futures margin account is dependent on the futures price changes. In addition, interest rate changes also affect the interest earnings. Indeed, if interest rate changes are positively correlated with futures price changes, then the futures position benefits from interest rate movements because as cash flows are received, more interest is earned. In this case, the risk of a futures position is reduced slightly. Conversely, if interest rate changes are negatively correlated with futures price changes, then the futures position suffers from interest rate movements. In this latter case, the risk is increased. This interest rate risk affects hedging performance because of the random cash flows received. This same risk is not present in a forward contract.

13.6

Spreadsheet Applications: Computing h

You can easily compute the optimal hedge ratio (h) by using a standard spreadsheet program such as Microsoft Excel. Next, we demonstrate this for Example 13.4.

EXAMPLE 13.4: Computing the Optimal Hedge Ratio (Solved Using Microsoft Excel)

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Generating the Inputs Consider the same data as in Example 13.3. Type in the numbers and the terms exactly as follows in an Excel spreadsheet (see Table 13.4): ■

Type Day t in cell A1, 0 in A2, 1 in A3, 2 in A4, and so on up to 15 in cell A17. You may speed up data entry by using the Auto Fill feature: type 0 in A2 and 1 in A3, highlight the cells A2 and A3, and then drag down the bottom right hand corner of the cursor so that the desired values get filled in cells A4 to A17.

■

Type S(t) in B1 and then fill out the Alloyum spot prices as given in Table 13.2: 1,215 in B2, 1,209 in B3, and so on, up to 1,194 in B17.

■

Type F(t) in C1 and then fill out the platinum futures prices F(t) as given in Table 13.2 for 16 consecutive trading days.

■

Type ΔS(t) in D1 and then type “= B3–B2” in D3, “= B4–B3” in D4, and so on. You may speed this up by using the Auto Fill feature: type “= B3–B2” in D3 and then drag down the bottom right hand corner of the cursor so that the desired values fill in cells D4 to D17.

■

Type ΔF(t) in E1 and then type “= C3–C2” in E3. Use Auto Fill as we have just mentioned and fill out the desired values in cells E4 to E17.

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TABLE 13.3: Computing h Using Microsoft Excel A

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Day t

B

C

D

E

S(t)

F(t)

ΔS(t)

ΔF(t)

F

G

H

ΔS(t) × ΔS(t) ΔF(t) ×ΔF(t)

ΔS(t) ×ΔF(t)

0

1,215 1,233

1

1,209 1,236

–6

3

36

9

–18

2

1,239 1,245

30

9

900

81

270

3

1,245 1,254

6

9

36

81

54

4

1,254 1,272

9

18

81

324

162

5

1,227 1,254

–27

–18

729

324

486

6

1,230 1,248

3

–6

9

36

–18

7

1,224 1,242

–6

–6

36

36

36

8

1,239 1,260

15

18

225

324

270

9

1,251 1,263

12

3

144

9

36

10

1,227 1,254

–24

–9

576

81

216

11

1,215 1,227

–12

–27

144

729

324

12

1,209 1,221

–6

–6

36

36

36

13

1,203 1,230

–6

9

36

81

–54

14

1,185 1,203

–18

–27

324

729

486

15

1,194 1,209

9

6

81

36

54

Sums

–21

–24

3,393

2,916

2,340

VARA

240.2571

205.5429

STDEV

15.5002

14.3368

COVAR

164.7429

CORREL

0.74138

h

0.8015

0.8015

SUMMARY

Computing Estimates from the Sample and the Minimum-Variance Hedge Ratio The entries in columns D and E are the two data series with which we work. You can do the computations as follows (and verify that these are the same as the sample variances that you computed in Example 13.3): ■

Type in a name for the sample variance such as vara in cell A20, type “= vara(d3:d17)” in D20 and hit return to get 240.2571, and type “= vara(e3:e17)” in E20 and hit return to get 205.5429.

■

Type stdev in A21, type “= stdev(D3:D17)” in D21 and hit return to get 15.50023, and type “= stdev(e3:e17)” in E21 and hit return to get 14.33677.

■

As Excel’s covariance formula is given for the population, you need to multiply the estimator by n/(n – 1) to get the sample estimate. Accordingly, type covar in A22, and type “= COVAR(D3:D17,E3:E17)*(15/14)” in D22 and hit return to get 164.7429.

■

Type correl in A23 and “= correl(d3:d17,e3:e17)” in D23 and hit return to get 0.74134.

■

Type h in A24 and “= d22/e20” in D24 and hit return to get 0.801501. This is the Equation 13.6 given in Result 13.1. You may also get h via the second result in Equation 13.6 by typing “= d23*d21/e21” in E24 and hitting return—it’s the same answer.

13.7

Summary

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1. Most futures contracts are traded to hedge spot commodity price risk. Producers may set up a long hedge (or a buying hedge) to fix the buying price for an input and a short hedge (or a selling hedge) to fix the selling price of an output. 2. As with any other derivatives, the costs and the benefits must be carefully weighed before hedging a spot commodity with futures. They involve direct and indirect costs: the brokers must be paid, and a trade may fetch a bad price. But hedging also has many potential benefits: it can stretch the marketing period, protect inventory value, permit forward pricing of products, reduce the risk of default and financial distress costs, and perhaps facilitate taking advantage of a tax loss or tax credit. 3. A perfect hedge completely eliminates price risk. It happens when (1) the futures is written on the commodity being hedged and (2) the contract matures when you are planning to lift the hedge. Generally, there is basis risk. The basis is defined as the difference between the spot price and the futures price. 4. For a seller of a futures contract who hedges output price risk, the effective selling price is (new basis + old futures price). For a buyer of a futures contract who hedges input price risk, the effective buying price is the negative of (new basis + old futures price). 5. We can use a statistical model, assuming past price patterns repeat themselves, to set up a risk-minimization hedge. To minimize the risk of a long portfolio of the spot commodity, sell short q = (n/f )h contracts, where n is the size of the spot position, f is the contract size, and h is the optimal hedge ratio (or

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minimum-variance hedge ratio) defined as covS,F /varF , where covS,F is the sample covariance between the change in the spot price (ΔS) and the change in the futures price (ΔF) and varF is the variance of the change in the futures price.

13.8

Appendix

Deriving the Minimum-Variance Hedge Ratio (h) Suppose you are planning to sell n units of a spot commodity at some future date and decide to hedge this exposure by selling short q futures contracts today. ■

Rewrite the change in the portfolio value between today (time 0) and some future date (time t) as ΔPortfolio = (n × Change in spot price) − (q × f × Change in futures price) = n [S (t) − S (0)] − q f [F (t) − F (0)] = nΔS − q fΔF

(13.8)

where f is the number of units of the futures contract, S(t) is the spot price at date t, F(t) is the futures price at date t, and Δ denotes a price change. ■

Select q so as to minimize the change in the portfolio value. First, use statistics to compute the variance of the portfolio’s price change:

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variance (aX − bY) ≡ a2 variance (X) + b2 variance (Y) − 2ab [covariance (X, Y)]

(13.9)

where a and b are constants and X and Y are random variables. As n, q, and f are constant parameters, set a = n, b = q f, X = ΔS, and Y = ΔF in Equation 13.9 to get 2

var (ΔPortfolio) = n2 vars + (q f) varF − 2nq f × covS,F

(13.10)

where varS is the variance of change in the spot price (ΔS), varF is the variance of change in the futures price (ΔF ), and covS,F is the sample covariance between the change in the spot and futures prices. ■

Using calculus, we minimize Equation 13.10 by taking the partial derivative with respect to q and setting it equal to zero: 𝜕 [var (ΔPortfolio)] = 2q f 2 varF − 2nf covS,F = 0 𝜕q The second partial derivative is a positive number, which indicates that this minimizes the equation. Rearrange terms to get the number of futures contracts to sell short (or go long in case you are setting up a buying hedge): q = (n/f) h where the optimal hedge ratio h equals (covS,F /varF ).

(13.11)

APPENDIX

Computing the Minimum-Variance Hedge Ratio (h) Example 13.4 shows how to input the values into a business calculator or a computer to determine the minimum variance hedge ratio (h). Here we show how to directly calculate h.

Statistical Approach Start by deciding how frequently and over what time interval you collect your data. Standard practice is to fix a time interval (daily, weekly, or monthly) and use the end of the interval’s settlement prices.

DATA SERIES ■

To cross-hedge Alloyum spot with platinum futures, we need their prices as basic inputs. These are reported in Table 13.5, whose first three columns are as follows: - The first column is labeled Day t, and it keeps track of the days. There are (T + 1) = 16 observations corresponding to sixteen consecutive trading days, where t “runs” from day 0 to day t = 15. - Columns 2 and 3 record the Alloyum spot price S(t) and the platinum futures price F(t), respectively, for these sixteen consecutive trading days. All prices are reported for one unit of the respective commodity.

PRICE DIFFERENCES

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■

Compute price differences for each price series. - Column 4 reports changes in the value for Alloyum spot. Denote the spot price change from date (t – 1) to date t as ΔS(t) ≡ S(t) – S(t – 1). Label column 4 as x(t) for convenience. For example, when t = 6, x (6) ≡ ΔS (6) − S (6) – S (5) = 1, 230 − 1, 227 = 3 - Column 5 does the same for platinum futures. Label this as y(t). For example, y (6) ≡ ΔF (6) = ΔF (t) ≡ F (t) − F (t − 1) = F (6) − F (5) = 1, 248 − 1, 254 = −6

■

Notice that when computing the “difference,” you lose the first observation. We now have T = 15 observations. We denote them by t, where t = 1, 2, . . . , 15.

■

We can now forget our second and third columns. We use the fourth and fifth columns to compute the minimum-variance hedge ratio h.

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TABLE 13.4: Price Data and Calculations for the Minimum-Variance Hedge Ratio (h) ΔS(t) ≡ S(t) – S(t – 1) ≡ x(t)

ΔF(t) ≡ F(t) – F(t – 1) ≡ y(t)

1,236

–6

3

1,239

1,245

30

3

1,245

1,254

4

1,254

5

[ΔF(t)]2 ≡ y(t)2

ΔS(t) × ΔF(t) ≡ x(t)y(t)

36

9

–18

9

900

81

270

6

9

36

81

54

1,272

9

18

81

324

162

1,227

1,254

–27

–18

729

324

486

6

1,230

1,248

3

–6

9

36

–18

7

1,224

1,242

–6

–6

36

36

36

8

1,239

1,260

15

18

225

324

270

9

1,251

1,263

12

3

144

9

36

10

1,227

1,254

–24

–9

576

81

216

11

1,215

1,227

–12

–27

144

729

324

12

1,209

1,221

–6

–6

36

36

36

13

1,203

1,230

–6

9

36

81

–54

14

1,185

1,203

–18

–27

324

729

486

15

1,194

1,209

9

6

81

36

54

Σx = – 21

Σy = – 24

Σx2 = 3,393

Σy2 = 2,916

Σxy = 2,340

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Day t

Alloyum Spot S(t)

Platinum Futures F(t)

0

1,215

1,233

1

1,209

2

Sums

[ΔS(t)]2 ≡ x(t)2

Some values useful for the computations follow: T

∑

x (t) = −21

∑

y (t) = −24

∑

x(t)2 = 3, 393

∑

y2 = 2, 916

∑

x (t) y (t) = 2, 340

t=1 15

t=1 15

t=1 15

t=1 15

t=1

APPENDIX

COMPUTATION-SIMPLIFYING TECHNIQUES - Three more columns are introduced to help simplify our calculations. - Columns 6 and 7 record the square of the price changes reported in Columns 4 and 5, respectively. Finally, Column 7 reports the product of the values in Columns 5 and 6. For example, when t = 6, we get [x(6)]2 = 32 = 9y (6) [ y(6) ] 2 = (−6)2 = 36 x (6) y (6) = −18

STATISTICAL ESTIMATES ■

To compute h from historical data, modify our formulas for covariance and variance by replacing the T with (T - 1) in the denominator to get an unbiased estimate.

■

We get the following parameter estimates to help us compute h (see Table 13.6): sample covariance, covS,F = 164.7429 sample variance of ∆S, varS = 240.2571 sample variance of ∆F, varF = 205.5429 sample standard deviations, sdS = 15.5002 and sdF = 14.3367 correlation coefficient between ΔS and ΔF, corrS,F = 0.7413

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■

These estimates are used to compute the minimum-variance hedge ratio h. The first part of Equation 13.7 of Result 13.1 gives h=

covS,F 164.7429 = = 0.8015 varF 205.5429

(13.12)

while the second part verifies the same result: h= ■

sdS 15.5002 0.7413 = 0.8015 corr = sdF S,F 14.3367

(13.13)

If you are estimating these values over a longer period, then you run the risk that a futures contract may mature and no longer trade. Suppose you are hedging with the “nearest maturity” futures contract, whose price changes are likely to have the best correlation with the spot price changes. Suppose this contract stops trading on Day 12. Select the futures contract that is going to mature next, get its futures prices for Days 12 and 13, and write the price difference ΔF(13) ≡ F(13) – F(12) in the first table in the cell corresponding to Day 13’s price difference.

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CHAPTER 13: FUTURES HEDGING

TABLE 13.6: Statistical Estimates and Computation of the Optimal Hedge Ratio h Statistical Estimate

Formula

Computations

Estimated Value

–21/15

–1.4000

T

a. Sample mean, x

1 x(t) T∑ t =1

b. Sample mean, y

1 y(t) 15 ∑ t =1

–24/15

–1.6000

1 [x(t) − x] × [y(t) − y] T−1 ∑ t =1

1 −21 −24 2, 340 − 15 ( 15 ) ( 15 )] 14 [

164.7429

15

T

c. Sample covariance of ΔS and ΔF, covS,F

15

=

1 x(t)y(t) − 15xy 14 {[∑ } ] t =1 T

d. Sample variance of ΔS, varS

1 [x(t) − x]2 T−1 ∑ t =1

240.2571

15

=

1 2 x(t)2 − 15x 14 [∑ ] t =1 T

2

1 [y(t) − y]2 T−1 ∑ t =1

1 −24 2, 916 − 15 ( 14 [ 15 ) ]

205.5429

f. Sample standard deviation of ΔS, sdS

+√varS

+√240.2571

15.5002

g. Sample standard deviation of Δ F, sdF

+√varF

+√205.5429

14.3367

covS,F

164.7429 15.5002 × 14.3367

0.7413

164.7429 205.5429

0.8015

e. Sample variance of ΔF, varF Copyright © 2018. World Scientific Publishing Company Pte. Limited. All rights reserved.

2

1 −21 3, 393 − 15 ( 14 [ 15 ) ]

h. Correlation coefficient between ΔS and ΔF, corrS,F i. Minimumvariance hedge ratio, h j. h (alternate computation)

sdS × sdF

covS,F varF sdS corrS,F sdF

0.7413

15.5002 ( 14.3367 )

0.8015

CASES

Econometric Approach: A Linear Regression Model Econometrics is the branch of economics that studies relationships between economic variables using mathematics and statistics. A staple of econometrics is the linear regression model,6 in which a dependent variable may be determined from the values of one or more independent variables, under the assumption that these variables are linearly related. In this case, the model takes the form ΔS (t) = 𝛼 + 𝛽ΔF (t) + u (t)

(13.14)

where 𝛼 and 𝛽 are unknown parameters and u(t) are independent and identically distributed error terms with an expected value of 0 and a constant variance. Next you have to estimate these parameters. The most popular approach is the least squares method, in which 𝛼 and 𝛽 are chosen so that the sum of squared errors Σt [ΔS(t) – 𝛼 – 𝛽ΔF(t)]2 is minimized. Interestingly, the value of 𝛽 obtained is identical to the hedge ratio h in Result 13.1. This makes life easy. Estimate the regression Equation 13.4 by hand or by using a standard statistical package like MATHEMATICA, MINITAB, SAS, or TSP, and the estimated value of 𝛽 will give you the hedge ratio. Spreadsheet programs like Microsoft Excel can also run regressions.

13.9

Cases

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Lufthansa: To Hedge or Not to Hedge . . . (Richard Ivey School of Business

Foundation Case 900N22-PDF-ENG-PDF-ENG, Harvard Business Publishing). A short case that explores the costs and benefits of hedging and the derivatives that may be used for hedging purposes. Enron Gas Services (Harvard Business School Case 294076-PDF-ENG). The case

considers the risks and opportunities of selling a variety of natural gas derivatives by a financial services subsidiary of the largest US integrated natural gas firm. Aspen Technology Inc.: Currency Hedging Review (Harvard

Business School Case 296027-PDF-ENG). The case examines how a small, young firm’s business strategy creates currency exposure and how one can manage such risks.

6

See standard econometrics textbooks like Amemiya (1994) or Johnston and DiNardo (1997) for a discussion of the linear regression model.

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13.10 Questions and Problems 13.1. Define a long hedge and a short hedge and give examples of each kind of

hedge. 13.2. What are the benefits of a corporation’s hedging? In your answer, explain

why the corporation and not the corporation’s equity holders must do the hedging. Are there any costs to a corporation’s hedging? 13.3. What is the difference between a perfect hedge and a cross hedge? Give

examples to clarify your answer. 13.4. If one cannot create a perfect hedge, what are the alternatives? Give an

example to explain your answer. 13.5. What is the basis for a futures contract? What happens to the basis on the

delivery date? 13.6. There is no futures contract on aviation fuel. Combo Air Inc. has to buy

3 million gallons of aviation fuel in three months. Suppose you are in charge of Combo Air’s hedging activities. You gather the following data:

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TABLE 13.7: Jet Fuel Cash Prices vs. Near Month Energy Futures Prices: Correlations of Price Changes 1986–88

1986

1987

1988

Heating oil futures

0.54

0.76

0.89

0.32

Gasoline futures

0.41

0.74

0.73

0.19

Crude oil futures

0.45

0.70

0.72

0.25

a. Which energy futures contract will you choose for hedging jet fuel

purchase? b. Is this a long hedge or a short hedge? 13.7. The variance of monthly changes in the spot price of live cattle is (in cents per

pound) 1.5. The variance of monthly changes in the futures price of live cattle for the April contract is 2. The correlation between these two price changes is 0.8. Today is March 11. The beef producer is committed to purchasing four hundred thousand pounds of live cattle on April 15. The producer wants to use the April cattle futures contract to hedge its risk. What strategy should the beef producer follow? (The contract size is forty thousand pounds.)

QUESTIONS AND PROBLEMS

13.8. Are hedging with forwards and futures contracts the same, or are there

different risks to be considered when using these two contracts? Explain your answer. 13.9. When you hedge a commodity’s price risk using a futures contract, give an

example where the counterparty is also hedging. Give an example where the counterparty is speculating. 13.10. Kellogg will buy 2 million bushels of oats in two months. Kellogg finds that

the ratio of the standard deviation of change in spot and futures prices over a two-month period for oats is 0.83 and the coefficient of correlation between the two-month change in price of oats and the two-month change in its futures price is 0.7. a Find the optimal hedge ratio for Kellogg. b How many contracts do they need to hedge their position? (The size of

each oats contract is five thousand bushels; oats trades in the CME Group.) 13.11. Explain why hedging is like buying an insurance policy. To buy an insurance

policy, you need to pay a premium; what is the corresponding premium in hedging? Give an example to clarify your answer. 13.12. Suppose that after you graduate, you plan to be a stock analyst for a major

financial institution. You know that if the stock market increases in value, you will get a job with a good salary. If the stock market declines, you will get a job, but the salary will be lower. How can you hedge your salary risk using futures contracts? Is this a perfect or imperfect hedge? Copyright © 2018. World Scientific Publishing Company Pte. Limited. All rights reserved.

13.13. Your company will buy tungsten for making electric light filaments in the

next three to six months. Suppose there are no futures on tungsten. How would you hedge this risk? (Discuss the type of hedge, general hedging approach, and guidelines that you would like to follow.) 13.14. You are the owner of a car rental business. If gasoline prices increase, your

car rental revenues will decline. How can you hedge your car rental revenue risk using futures contracts? Is this a perfect or imperfect hedge? 13.15. What commodity price risk does Southwest Airlines hedge, and why? Has it

always been successful in its hedging program? 13.16. What is risk-minimizing hedging? Briefly outline how you would set up a

risk-minimizing hedge. Is a risk-minimizing hedge a perfect or imperfect hedge? Explain your answer.

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13.17. Canadian American Gold Inc. (CAG) has half its gold production from

mines located in Canada, while the other half is from those in the United States. CAG uses a quarter of its production for making gold jewelry sold at a fixed price through stores in the two nations, and the rest is sold on the world market, where the gold price is determined in US dollars. Canadian profits are repatriated to the United States, where CAG’s headquarters are located. CAG’s chief executive officer wants to use futures contracts to hedge the entire production of ten thousand ounces of gold and as many other transactions as possible. He communicates his desire to you but seeks your opinion one last time before the orders go out. Devise a sensible hedging strategy that would still be in line with the CEO’s wishes (assume x is the quantity used for making gold jewelry in the United States). 13.18. The spot price of gold today is $1,505 per ounce, and the futures price for a

contract maturing in seven months is $1,548 per ounce. Suppose CAG puts on a futures hedge today and lifts the hedge after five months. What is the futures price five months from now? Assume a zero basis in your answer. 13.19. Suppose that Jewelry Company is planning to sell twenty thousand ounces of

platinum at some future date. The standard deviation of changes in the futures price per ounce sdF is 12.86, that for changes in the spot price per ounce sdS is 14.38, and the correlation coefficient between the spot and futures price changes corrS,F is 0.80. a. Compute the optimal hedge ratio for Jewelry Co. b. How many contracts do they need to hedge their position? (The contract Copyright © 2018. World Scientific Publishing Company Pte. Limited. All rights reserved.

size is fifty ounces.) c. Will this be a buying or a selling hedge?

QUESTIONS AND PROBLEMS

13.20. (Microsoft Excel) Given the following data, compute the hedge ratio for a

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risk minimizing hedge. Day t

Alloyum Spot S(t)

Platinum Futures F(t)

0

1,233

1,245

1

1,219

1,256

2

1,118

1,130

3

1,246

1,264

4

1,250

1,280

5

1,219

1,223

6

1,230

1,248

7

1,227

1,280

8

1,249

1,260

9

1,225

1,289

10

1,227

1,254

11

1,223

1,255

12

1,211

1,223

13

1,203

1,267

14

1,189

1,213

15

1,199

1,219

285

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III

Options

CHAPTER 14

Options Markets and Trading

CHAPTER 15

Option Trading Strategies

CHAPTER 16

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Option Relations

CHAPTER 19

The Black–Scholes– Merton Model

CHAPTER 20

Using the Black– Scholes–Merton Model

CHAPTER 17

Single-Period Binomial Model

CHAPTER 18

Multiperiod Binomial Model

14 Options Markets and Trading 14.1 Introduction Copyright © 2018. World Scientific Publishing Company Pte. Limited. All rights reserved.

14.2 Exchange-Traded Options 14.3 A History of Options Early Trading of Options The Year 1973: The Watershed Year and After EXTENSION 14.1 Options on Futures

14.4 Option Contract Features Maturity Dates Strike Prices Margin Requirements and Position Limits

Dividends and Stock Splits

14.5 Options Trading, Exercising, and the Expiration Process The Trading Process Exercising the Option The Expiration Process

14.6 Options Price Quotes 14.7 Regulation and Manipulation in Options Markets 14.8 Summary 14.9 Cases 14.10 Questions and Problems

EXCHANGE-TRADED OPTIONS

14.1

Introduction

Believe it or not, governments trade derivatives. Distinguished economist Lawrence Summers recounted an interesting experience when he was deputy secretary of the US Treasury (see Summers 1999, p. 3):

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Any doubt I might have had about the globalization of economic thinking was shattered when I met with Chinese Premier Zhu Rhongji in early 1997 in the same pavilion where Chairman Mao had received foreign visitors. After being offered a Diet Coke, I was asked a variety of searching questions about the possible use of put options [emphasis added] in defending a currency, and how they might best be structured. A communist leader dabbling in derivatives! A more recent example of derivative securities being used to support a government’s fiscal and monetary policy occurred in the United States during the 2007 credit crisis. To reduce the cost of the capital infusion by the US Treasury into financial institutions under the Troubled Asset Relief Program (TARP), warrants—a type of call option—were used. These warrants, issued to the US Treasury by the banks receiving the TARP funds, provided additional upside potential to the US Treasury. After the banks recovered, the US Treasury made significant profits on its warrant positions, reducing the cost of the TARP program. Exactly as hoped! This chapter continues our discussion of options and their markets started in Chapter 5. First, we review the history of options trading, which is divided into two phases around the watershed year of 1973. Next we describe some features of options contracts. We end the chapter with regulatory issues, including market manipulation. Although our discussions primarily revolve around equity options, many of the features discussed here are similar to other options encountered later in this book and elsewhere in life.

14.2

Exchange-Traded Options

Why trade options? Unlike futures, options have asymmetric payoffs, a special feature that allows users to carve out richer payoff structures than what’s otherwise obtainable. This makes options valuable tools for hedging, speculation, and risk management. Because the holder is rewarded when the underlying moves favorably but has no loss when it moves adversely, options are similar to insurance contracts and require a premium payment at their onset. But regular use of options can get expensive, and one must carefully weigh the costs and benefits before trading an option. We skip discussing the applications and uses of options because of its similarity with our earlier discussion of the applications and uses of derivatives (see section 1.3), swaps (see section 7.5), and forwards and futures (see section 8.2). Although options have traded over the counter (OTC) for hundreds of years, the market really took off when exchange-traded options were introduced on the

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Chicago Board Options Exchange (CBOE) in 1973. The reasons are twofold. First, an exchange creates a central marketplace: an orderly, efficient, and liquid market with low transaction costs where traders can enter or exit positions with ease. Second, a clearinghouse for the exchange guarantees contract performance and makes trades virtually free from credit risk. The resulting popularity of plain vanilla exchangetraded options paved the way for the development of the now gigantic OTC market for trading customized, complex options. The causation did not happen in reverse. Although OTC options have traded for centuries, their market remained small and inefficient, fraught with credit and legal risks, never catching on with the investing public until after exchange traded options became popular. Why did it happen this way? To find out, read on.

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14.3

A History of Options

“History has many cunning passages, contrived corridors,” wrote T. S. Eliot in the poem “Gerontion.” This describes the history of options. Option-like contracts arise naturally in economic life. One enters a call-like agreement when you pay a deposit in advance of buying a house with the provision that your failure to complete the purchase forfeits this amount; you enter into a put-like contract when you buy insurance by paying a premium. The term premium also denotes the price of an option, which reinforces the close link between options and insurance. Despite these similarities, the options and insurance markets went their separate ways. Insurance contracts have enjoyed popular support for centuries, but options’ acceptance has waxed and waned over the years. Options and futures traded side by side in Amsterdam during the Dutch golden age of the seventeenth century. They have traded for over two hundred years in small OTC markets in London, New York, and several cities of Continental Europe. Options traded as bilateral contracts that were loaded with credit, legal, and liquidity risks. They were viewed as tools for speculation that served no worthwhile economic purpose. In fact, the US Securities and Exchange Commission (SEC; created after the great stock market crash of 1929) repeatedly tried to outlaw options. The market ebbed and flowed depending on the participants. It expanded when big financiers wrote options because their reputation and deep pockets assuaged the buyers about credit risk. The opening of the CBOE in 1973, support from academic economists, and the Black–Scholes–Merton pricing model helped options reach the mainstream and remove their stigma of gambling. The information technology revolution of the 1990s ushered in the era of electronic exchanges and web-based trading. Insurance and options markets have converged again. Nowadays options are viewed as risk management tools that “complete the markets” by allowing traders to carve out different portfolio payoffs. They trade on many exchanges around the globe, and traders can transact through a single electronic platform.

A HISTORY OF OPTIONS

Despite this convergence, the history of options trading is sufficiently different from that of futures trading to merit further discussion. We divide the history into two phases: 1. Early trading of options. Option-like contracts have traded since antiquity, with options trading in organized OTC markets in Amsterdam, London, and New York City in past centuries. 2. The year 1973: the watershed year and after. The Nobel Prize–winning Black– Scholes–Merton model gave traders a vital tool for pricing and hedging options. The opening of the CBOE ended the old, inefficient OTC markets and brought numerous innovations. Post-1973 saw the development of a variety of contracts, the opening of new exchanges, the automation of trading, and the consolidation and linkages among exchanges. This story is akin to the history of futures trading discussed in Chapter 8. That’s hardly surprising because economic, political, regulatory, and technological changes similarly affected the options markets.

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Early Trading of Options Options and futures traded in Western Europe over four centuries ago. Before exploring that history, however, we mention that the first recorded option usage occurs in a story telling how Thales the Milesian (624–545 BC) purchased an option to buy the use of olive presses in the next harvest year, which he was sure would yield a bountiful harvest. His prediction was correct, and he profited handsomely from exercising his option.1 Let us move forward two millennia, and arrive in the Netherlands. Gelderblom and Jonker (2005) tell the story of how sophisticated financial markets developed in Western Europe, which, nonetheless, were vexed by problems that continue to afflict us to this day (see Table 14.1 for a timeline of options trading). Records from Amsterdam document how grain dealers in the city used both futures and options in the 1550s and the following decades. Fearing that speculators could use derivatives to manipulate staple food prices and cause social unrest, the authorities repeatedly banned forward trading. However, the market did not disappear; it simply went underground and was confined to the insiders. Subsequently, the market again became public, expanding to include forwards, futures, and call and put options on equities. The 1630s saw an extraordinary speculative frenzy over rare tulip bulbs, with futures and options helping hedgers and the speculating public. We discuss this episode later in this chapter. A small options market for sophisticated traders continued to operate in Continental Europe, London, and New York for over two centuries (see Malkiel and Quandt 1969). For example, during the 1690s, a well-organized options market existed in London, and it continued to exist despite the ban on options and futures promulgated under Barnard’s Act of 1733. In 1821, the London Stock Exchange Committee proposed a rule that would forbid members from dealing in options. Loud protest came from a large number of members; they even raised money to build a rival stock exchange building. The ban was never implemented, option trading continued 1

This story is from Jowett’s translation of Chapter 11 of Book 1 of Aristotle’s Politics (etext.library. adelaide.edu.au/a/aristotle/a8po/).

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TABLE 14.1: Early History of Options Trading Year

Development

600 bc onward

Thales of Miletus traded optionlike contracts

1550–1650

Birth of options and futures trading in Amsterdam

1630s

Futures and options traded on tulips in the Netherlands

1860–1900

Russell Sage extensively traded calls, puts, spreads, and straddles

1910

The Put and Call Brokers and Dealers Association was founded

(restricted but) uninterrupted, and Barnard’s Act itself was repealed in 1860. The European options market remained small because many prominent firms refused to participate. A New York City options market became active during the last four decades of the nineteenth century, when Russell Sage operated as a financier, securities trader, options writer, and broker on a vast scale. Russell Sage’s financing and trading activities relate to several important features of option markets:

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1. The importance of collateral. An article titled “Russell Sage” (New York Times, August 5, 1903), attributed Sage’s success to his being “an excellent judge of collateral.” It noted that “crises and panics have passed him unharmed, and when others were eager to borrow he was as eager to lend—if the collateral was satisfactory.” 2. The invention of straddles and spreads. Sarnoff (1965) gave Sage credit for inventing the spread and the straddle and suggested that his extensive dealing in calls and puts as well as these two creations led to the development of the “modern system of selling stock-option contracts.” These activities earned him the nickname “Old Straddle.” 3. Dividend adjustments. A dividend lowers the value of an asset; consequently, a dividend lowers the value of a call and raises the price of a put. A client once took the aged Sage to court in a dispute about dividend adjustments and an option’s value (see “Dividends and Puts,” New York Times, April 6, 1887). Today options exchanges have well-defined policies for these adjustments (see section 14.4 and Example 14.4 for further discussions). 4. Put–call parity. Sage devised conversions, where he created synthetic loans by using the put–call parity principle and effectively charged interest rates higher than what the usury laws would allow. The OTC option market in the nineteenth and the early twentieth centuries wasn’t too different from that in historic Amsterdam. Options contracts were written in printed forms where traders filled in the details and had them notarized. The dealers sold options by word of mouth or by advertising in the newspapers. The Put and Call Brokers and Dealers Association was established in 1910 to organize options trading in the United States.

A HISTORY OF OPTIONS

The government continued to view options trading with suspicion for good reasons. Particularly in the early twentieth century, speculators would often use options to manipulate stock prices. After the stock market crash of 1929, the US Congress sought to ban options trading, but did not do so. Options trading volume (in terms of shares), which was about half of 1 percent of the New York Stock Exchange (NYSE) stock trading volume in 1937, rose to around 1 percent in 1955 and hovered around that number during the 1960s; however, trading remained anemic, and the stigma of gambling prevented the options market from enjoying vigorous growth.

The Year 1973: The Watershed Year and After

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This landscape began to change in 1973 with the opening of the CBOE and the publication of the Black–Scholes–Merton options pricing model.

THE OPENING OF THE CHICAGO BOARD OPTIONS EXCHANGE The CBOE opened its doors on April 26, 1973. Just 911 call options contracts traded on sixteen stocks on the inaugural day. The trading floor was set up in the previous Chicago Board of Trade members’ smoking lounge. The daily trading volume at the end of the first month exceeded the average daily volume in the old, OTC options market. Despite its ups and downs, a remarkable expansion of options markets followed the opening of the CBOE—see Table 14.2 for some milestones in this extraordinary march. The mid-1970s saw the introduction of options trading on several US exchanges and in other nations. In 1975, the clearing facility of the CBOE was reorganized as the Options Clearing Corporation (OCC), which functions as the common clearinghouse for most exchange-traded listed options in the United States and has virtually eliminated credit risk. The same year, computerized price reporting was also introduced in the CBOE. During the three-year period 1977–80, the SEC put a moratorium on additional listings pending a review of the options industry’s growth. The stock market crash in October 1987 led to a temporary waning of interest in options. Still the innovations continued, and a variety of new contracts were introduced. The IT revolution of the 1990s led to several key developments: automation lowered transaction costs, many exchanges introduced electronic trading, greater links were established among exchanges, and a wave of mergers and consolidations transformed the markets. A VARIETY OF OPTIONS Most options varieties were introduced during the 1970s and 1980s. In 1977, put options started trading at the CBOE. The year 1982 saw the beginning of currency options trading on the Philadelphia Stock Exchange and Treasury bond futures options on the Chicago Board of Trade, which brought futures exchanges into the options business (see Extension 14.1). The following year, cash-settled options on broad-based stock indexes (including options on the Standard and Poor’s [S&P] 500 index) were launched at the CBOE. As with futures, cash settlement allowed the creation of new kinds of options with rich payoff structures.

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TABLE 14.2: Some Milestones in Options Markets, 1973 and After

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Year

Development

1973

First options exchange, the Chicago Board Options Exchange (CBOE), was founded, with call options trading on sixteen underlying stocks

1975

Call options began trading in the American Stock Exchange, the Philadelphia Stock Exchange, and the Montreal Stock Exchange in Canada; the Options Clearing Corporation (OCC) was established for exchange-traded options in the United States

1976

Calls began trading on the Pacific Exchange and the Australian Options Market

1977

Put options were introduced at the CBOE

1977–80

The Securities and Exchange Commission put a moratorium on the options market

1978

The European Options Exchange and London Traded Options Market were established

1982

Currency options started trading at the Philadelphia Stock Exchange

1982

Options on US Treasury bond futures started trading at the CBOT

1983

Options on broad-based stock indexes started trading at the CBOE

1987

Stock Market crash in October 1987 set back the options market, from which it took years to recover

1993

Introduction of the CBOE Volatility Index (VIX), a key measure of market expectations of near-term volatility conveyed by Standard and Poor’s index options prices

1998

Deutsche Borse AG and Swiss Exchange started Eurex, a joint platform for trading and clearing derivatives trade

2000

Euronext was founded through the merger and consolidation of several European exchanges

2000

First all-electronic exchange in the United States, the International Securities Exchange, was founded

By mid-1983, exchange-traded options were available for hedging and speculating on equity, commodity, currency, interest rates, and stock indexes. Many options were subsequently introduced in these categories. For example, the CBOE introduced long-lived equity options, LEAPS (Long-Term Equity AnticiPation Securities), with up to three years’ life in 1990, options on sector indexes in 1992, FLEX options (which allow traders to modify some key specifications) in 1993, and options on country indexes in 1994. On the basis of academic research by Professor Robert Whaley, the CBOE unveiled in 1993 the VIX (CBOE Volatility Index), which measures the market’s estimate of near-term volatility as implied by the S&P 500 stock index option prices. The VIX index was modified in 2003 from measuring an implied volatility to that of an expected volatility. Colloquially known as the “Fear Gauge,” VIX is a widely used barometer of investor sentiment and market volatility.

A HISTORY OF OPTIONS

295

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The CBOE introduced futures contracts on VIX in 2004 and option contracts on VIX in 2006—we discuss the VIX in chapter 20.

NEW EXCHANGES AND THE AUTOMATION OF TRADING Many existing exchanges embraced options trading, and new options exchanges were formed. In 1975, call options commenced trading at the American Stock Exchange, the Philadelphia Stock Exchange, and the Montreal Stock Exchange in Canada. In 1976, they were introduced on the Pacific Exchange in San Francisco and in the Australian Options Market, which became the first options market outside North America. In 1978, two of the world’s oldest exchanges introduced options trading in Europe: the European Options Exchange was founded by the members of the Amsterdam Stock Exchange, and the London Traded Options Market was established as a part of the London Stock Exchange. By the end of the decade, BOVESPA became the first exchange to introduce options trading in Brazil. During the 1980s, the exchanges worked on expanding the market and jockeyed for greater market share. The global options market suffered a severe setback during the stock market crash of October 1987, and it took over a decade for the volume to return to pre-crash levels. As the 1990s progressed, several discernible patterns emerged. Competition put relentless pressure to cut costs, and IT advances made that possible. Exchanges dismantled floor trading or began as completely electronic exchanges. With their commitment to floor trading, US exchanges kept falling behind. Wake-up calls came when Eurex and Euronext were founded in 1998 and 2000, respectively. Established in 2000 as an all-electronic options exchange, the International Securities Exchange (now part of Nasdaq Inc.) became the world’s largest equity options exchange. Stiff competition came from the OTC derivatives markets, which grew faster than the exchanges. Fed chairman Alan Greenspan (recall Chapter 10) used this fact to justify less regulation and a hands-off policy toward the OTC markets. The US exchanges regrouped and rethought their options (no pun intended!) and strategies. Seeing that electronic trading was the wave of the future for trading plain vanilla securities, they acquired or merged with organizations that focused on electronic trading or developed their own automated trading systems.

EXTENSION 14.1: Options on Futures Why trade options on futures? Recall the observation from Chapter 1 that trading gravitates to the most liquid markets with minimum transaction costs. As the stock market is very liquid with minimal transaction costs, it’s no surprise that we have bustling markets for equity options. But storage costs and convenience yields for many commodities in the spot market make it easier for traders to hedge and speculate on commodities in the more liquid futures markets. Consequently, it’s natural for options on futures to trade on such commodities. For options on spot (also called spot options), the spot commodity is obtained when the option is exercised. By contrast, exercise of a futures option gives the holder a position in the underlying futures. In the United

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States, active futures as well as futures options markets exist for diverse commodities such as cattle, cotton, corn, crude oil, gold, and silver. These options trade on the same exchange where the futures contracts trade. Simultaneous trading of futures and futures options helps the price discovery process. The example below shows how a futures option works. Notice that both spot options and futures options behave similarly in response to a movement in the underlying.

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EXT. 14.1 EX. 1: Call and Put Options on Futures ■

Today is January 1. Suppose the April gold futures price is $1,515 per ounce. Futures options with a contract size of one hundred ounces and for $1,490, $1,495, $1,500, $1,510, $1,520, $1,530, and several other strike prices (per ounce) trade in the New York Mercantile Exchange of the CME Group.

■

Consider a call option on a commodity futures. If you exercise this option, you will receive a long position in the underlying futures contract and a cash payment equal to the excess of the futures price over the strike price. For example, if you exercise an April 1,500 call, you end up long an April futures and receive a payment of (1,515 − 1,500) = $15 per ounce, $1,500 in all. If the payment is negative, do not exercise—enter the futures at zero cost in the market instead.

■

The exercise of a put option on a commodity futures puts the holder into a short position in the underlying futures and brings in a cash amount equal to the strike price minus the futures settlement price. For example, if you exercise an April 1,520 put, you end up being short April futures and receive a payment of (1,520 − 1,515) = $5 per ounce.

■

Obviously, the long has to pay premiums for these options. A modified version of the Black–Scholes–Merton model can be used to price futures options. More advanced models like the Heath–Jarrow–Morton model (developed in 1987, published in 1992) can also be used for this purpose (see Amin and Jarrow 1991).

Notice that if you replace the futures price F(T) with the stock price S(T), then futures options payoffs are similar to equity options payoffs. For example, consider a call futures option and a regular equity call option on a stock with the same strike price K. If you exercise the equity call, your payoff is [S(T) − K]. If you exercise the call futures option, you receive [F (T) − K] and a long futures position on the stock. As the futures contract has a zero value, it can be liquidated at no cost. Exercise of a put futures option leads to a short position in the underlying futures plus the strike minus the futures price, [K − F (T)], and a similar argument holds.

14.4

Option Contract Features

Let us examine the contract specifications for CBOE equity options.2 The CBOE regular trading session for options is from 8:30 am to 3:00 pm Central Time (Chicago time). CBOE equity options are American, and they can be exercised any time from purchase until the option’s last trading day, which is the third Friday of the expiration month. If the third Friday is a legal holiday, then the option expires the day before. Each option is for one hundred shares of the underlying stock. If exercised, the stock share is delivered three business days later. 2

See www.cboe.com.

OPTION CONTRACT FEATURES

The minimal tick size is $0.05 for options trading below $3 and $0.10 for options trading above. However, options series are quoted in pennies ($0.01) for options prices below $3 and in nickels ($0.05) for higher options prices.

Maturity Dates For a particular stock, options belong to an assigned quarterly cycle, either the January, February, or March cycle. With respect to a particular cycle, options maturities are listed for “two near-term months plus two additional months from the January, February or March quarterly cycles (www.cboe.com).” An example explains this procedure.

EXAMPLE 14.1: Determining the Expiration Months ■

The January cycle has the months January, April, July, and October. The February cycle has February, May, and so on. Each stock approved for options trading belongs to one of the cycles. For example, IBM has a January cycle, Hewlett Packard has a February cycle, and Walmart Stores has a March cycle.

■

Consider the expiration months for traded options on IBM starting January 1. - On January 1, IBM has options expiring in January, February (the two near-term months), April, and July (the next two months from the quarterly cycle). - After the third Friday of January, one can trade IBM options maturing in February, March, April, and July. - After the third Friday of February, one can trade options maturing in March, April, July, and October.

Copyright © 2018. World Scientific Publishing Company Pte. Limited. All rights reserved.

■

Different kinds of options have different expiration dates. For example, LEAPS are issued with threeyear maturities and expire on the third Friday of the expiration year. Quarterly options (or the “quarterlies”), which can be written on some indexes and exchange-traded funds, expire on the last business day of each calendar quarter. Weeklys, which can be traded on many stocks, ETFs, and indexes, typically expire on Friday of each week.

Strike Prices When an option with a particular maturity starts trading, strike prices for in-, at-, and out-of-the-money are listed. If the stock price is over $200, the strikes are issued $10 apart. Between $200 and $25, the strikes are issued $5 apart, and below $25, only $2.50 apart. The CBOE has also introduced $1 strike price intervals and $2.50 strike price intervals, under certain terms and conditions, for many actively traded stocks. Strike prices are adjusted for stock dividends and stock splits but not for cash dividends. If the stock price moves significantly before the option expires, then new strikes are listed to maintain the balance with respect to in-, at-, and out-of-themoney. The next example illustrates this process.

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EXAMPLE 14.2: Determining the Strike Prices ■

Suppose the stock opens at $21 on Monday following the third Friday of the month, when new long-maturity options are listed. This is the first trading day after the shortest maturity option expires. Newly listed options (both calls and puts) have strike prices of $20 and $22.50, immediately above and below the current stock price. Additional in- and out-of-the-money strikes are also listed at this time.

■

Next, suppose that only the $20 and $22.50 options are listed, and the stock rallies. When it reaches $22.51, options with a strike price of $25 are introduced. If it goes beyond $25, options with the next higher strike of $30 start trading, and so forth. Of course, one can still trade options with strike prices $20 and $22.50, which were listed earlier, until these contracts expire.

■

Liquidity is usually the greatest for options with strike prices near the current stock price. Trading volume and the number of contracts outstanding tend to decline as the stock goes deeper in- or out-of-the-money.

Margin Requirements and Position Limits

Copyright © 2018. World Scientific Publishing Company Pte. Limited. All rights reserved.

The margin (borrowing) restrictions for long positions in options can be summarized as follows: (1) short-term options (nine months or less) must post the full purchase price with zero borrowing and (2) long-term options (more than nine months) can borrow up to 25 percent of the purchase price. Options sellers are usually required to keep the full purchase price plus some extra cash as a cushion against risk. An example illustrates these restrictions.

EXAMPLE 14.3: Margin Account Adjustments ■

Long call or long put (maturing in less than nine months). For example, from Figure 14.1, if Ms. Long is buying five IBM March 2009 (IBMCR) calls worth $2.30 per share, she will need to keep (Ask price × Number of contracts × Contract size) = 2.30 × 5 × 100 = $1,150 in her margin account.

■

Long call or long put (maturing in over nine months). The initial (maintenance) margin requirement is 75 percent of the cost (market value). For example, if Ms. Long wants to buy ten IBM January 2010 (WIBAR) contracts worth $12.50, she will have to keep at least 75 percent of 12.50 × 10 × 100 = 0.75 × 12,500 = $9,375 of her own money as margin and borrow up to $3,125 from her broker.

■

Options writer. Suppose Mr. Short is selling ten IBM October 95 (IBMVS) put options (see Figure 14.1). Then margin would be computed as follows: - 100 percent of contract proceeds (13.30 × 10 × 100) $13,300 - Plus 20 percent of aggregate contract value (0.2 × 90.42 × 1,000) 18,084 - TOTAL amount (13,300 + 18,084) $31,384 Also compute - Put option proceeds (13.30 × 10 × 100)

$13,300

OPTION CONTRACT FEATURES

- Plus 10 percent of aggregate exercise price (0.1 × 95 × 1,000) - TOTAL amount (13,300 + 9,500)

9,500 $22,800

The margin Short has to keep is $31,384, which is the larger of the two.

If the options holder-writer has positions in other options or the underlying, the CBOE adjusts these margin restrictions, taking into account the portfolio’s overall risk. For example, the margin required for a long stock plus a long put position is significantly lower than a long put position alone. The adjusted margin takes into account the reduced risk from the long stock hedging the long put position. Options position limits are imposed to reduce the potential for market manipulation. For actively traded stocks with large capitalizations (e. g., IBM or Microsoft), the position limit is 250,000 contracts. Stocks with smaller capitalizations have smaller position limits. Exemptions may be granted for qualified hedging strategies.

Dividends and Stock Splits

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Options are not adjusted for cash dividends or stock distributions that do not exceed 10 percent of the stock’s value. When a stock splits or pays large stock dividends, the option’s strike price is adjusted on the ex-dividend day or the ex-split day. The adjustment depends on whether the split or stock dividend is an integer multiple such as 3 for 1 or a fractional such as 3 for 2. The OCC has developed these rules and procedures in an attempt to guarantee that options traders do not suffer losses from these events.

EXAMPLE 14.4: Options Adjustment for Stock Splits and Stock Dividends IBM’s stock price is $90.42 (see Figure 14.1). Consider the April 90 call contract.

An Integer Multiple split ■

Suppose IBM has a 3 for 1 split. An investor who held one share would now have three shares worth $30.14 per share. On the ex-distribution day, the number of options will increase by a factor of 3, and the strike price will be reduced by a factor of 3.

■

The aggregate exercise price before the split was 100 × $90 = $9,000. After the split, it equals 3 × 100 × $30 = $9,000. They are the same—the adjustment makes the option neutral with respect to a stock split.

A Fractional split ■

Suppose IBM has a 3 for 2 split. An investor holding two shares will now own three shares. To prevent arbitrage, the new share’s price will be 90.42/1.5 = $60.28, where 3/2 = 1.5 is the split ratio.

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■

In this case, instead of adjusting the number of contracts, the number of shares in each contract is multiplied by the split ratio: 100 × 1.5 = 150. The new strike price is obtained by dividing the old strike price by the split ratio. Thus the new exercise price after the split is 90/1.5 = $60.

■

Notice that the aggregate exercise price before the split was $9,000. After the split, the aggregate exercise price is 150 × $60 = $9,000. Again, this adjustment makes the option neutral with respect to the stock split.

Stock Dividends ■

Large stock dividends (greater than 10 percent of the stock’s price) are called stock splits and are treated accordingly. For example, a 3:1 split is really a 200 percent stock dividend, and a 3:2 split is a 50 percent stock dividend.

Reverse Split ■

If IBM has a 1 for 2 reverse split, the stock price would become $180.84. The contract is adjusted so that each original option is now an IBM April 180 call covering fifty shares.

14.5

Options Trading, Exercising, and the Expiration Process

Copyright © 2018. World Scientific Publishing Company Pte. Limited. All rights reserved.

The Trading Process Options trading involves the three-step process of execution, clearing, and settlement inherent to exchange-traded securities. As with futures, only a clearing member can clear trades. A clearing member must be a member of an options exchange, satisfy OCC’s minimum capital, and meet other requirements. He must maintain efficient operations on the trading floor and in the back office. Clearing members pay both the exchange and the OCC a fee for each traded contract. They recoup these expenses from the fees charged to their customers.

Exercising the Option The process begins when the holder conveys her intention to exercise the option to her broker. To ensure exercise on a particular day, the notification must be given before the broker’s daily cutoff time. Next, the broker submits an exercise notice to the OCC. The OCC sends the notice to one or more clearing members holding short positions on the same contract. The clearing member, in turn, assigns it (either randomly or on a first-in, first-out basis) to one or more customers who hold short positions in the contract. The clearing member representing the short is obligated to sell (in the case of a call) or buy (in the case of a put) the underlying shares at the specified strike price. The OCC then arranges for the delivery of the shares of stock and the exchange of strike price funds.

OPTIONS PRICE QUOTES

The Expiration Process The expiration date is the last day an option trades. For listed stock options, this is the third Friday of the expiration month. Brokerage firms must submit exercise notices to the OCC. The OCC has developed a procedure known as exercise by exception (also called ex-by-ex) to expedite the handling of the exercise of expiring options. Every contract in-the-money will be automatically exercised unless the clearing member advises OCC otherwise. Currently equity and index options that are in-the-money by at least 1 cent are automatically exercised. Nonetheless, most brokers still require their customers to notify their intention to exercise even if an option is in-the-money. When an equity option is exercised or assigned and it is not covered by stock holdings in the account, sufficient purchasing power must be in the account to make or take delivery of the shares. Otherwise, the broker may close your expiring options positions, even without notification, on the option’s last trading day. If the expiring position is not closed and there is insufficient purchasing power, the broker may submit “do not exercise” instructions to the clearing member without notification.

Copyright © 2018. World Scientific Publishing Company Pte. Limited. All rights reserved.

14.6

Options Price Quotes

Wire news services, websites, and many major daily newspapers carry options price quotes. Because each stock has calls and puts that differ in terms of exercise price and expiration months, an exchange typically has tens of thousands of options trading. All options contracts of the same type written on the same underlying stock belong to the same option class. For example, different IBM calls belong to one option class, whereas different IBM puts belong to another. Within a particular class, all options with the same strike price and expiration date belong to an option series; for example, IBM March 100 calls belong to one option series, IBM April 90 calls belong to another option series, and IBM April 100 puts belong to yet another option series. Example 14.5 illustrates IBM’s options price quotes.

EXAMPLE 14.5: IBM Options Price Quotes ■

IBM’s stock price is $90.42, which is up $1.80 (2.03 percent) from the previous day’s close. Figure 14.1 gives the IBM options “chain table” data from March 12, 2009, compiled from the CBOE website (the figure has been condensed and all quotes are twenty minutes delayed). Calls are written on the left side and puts are written on the right side of the figure, with the strike price in the middle. The first set of numbers reports call and put prices for March 2009, with the shaded region reporting options that are in-the-money.

■

Consider the tenth entry under “Calls,” for which $90 is the strike price. The row contains the following data, moving from left to right: - This option is also identified by IBMCR—C stands for the third month, March, and R stands for the strike $90.

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-

The price $2.27 is recorded under “Last Trade,” which stands for an option’s last reported trade price. The $0.56 denotes the price change from the previous settlement price. The bid price $2.25 and the ask price $2.30 are reported per option on one IBM share. The 6,516 under “Volume” is the number of contracts that have traded so far during the trading day. “Interest” 14,198 is the open interest, which is the total number of outstanding contracts.

FIGURE 14.1: IBM Options Quote from CBOE Website (IBM) International Bus Mach Corp Com March 2009 Calls

Copyright © 2018. World Scientific Publishing Company Pte. Limited. All rights reserved.

Symbol

Puts

Last Change Bid Ask Volume Interest Strike Symbol Last Change Bid Ask Volume Interest Trade Price Trade

IBMCX 43.40

2.40

45.20 45.60

n.a.

n.a.

45.00 IBMOX

0.10

0.00

0.05 0.05

35

123

IBMCU 36.00

−6.60

40.30 40.60

n.a.

n.a.

50.00 IBMOU 0.05

0.00

0.05 0.05

148

2,350

IBMCV 33.40

4.40

35.20 35.60

n.a.

5

55.00 IBMOV 0.05

0.00

0.05 0.05

7

325

IBMCL 25.20

1.20

30.20 30.60

7

92

60.00 IBMOL

0.05

0.00

0.05 0.05

7

827

IBMCM 20.90

−3.00

25.30 25.60

4

117

65.00 IBMOM 0.05

0.00

0.05 0.05

34

1,309

IBMCN 19.60

0.60

20.30 20.60

14

218

70.00 IBMON 0.05

0.00

0.05 0.10

20

3,153

IBMCO 12.80

−1.70

15.40 15.60

30

578

75.00 IBMOO 0.09

−0.01

0.05 0.10

307

7,409

IBMCP 10.45

1.35

10.50 10.70

116

2,874

80.00 IBMOP 0.20

−0.05

0.15 0.20

828

10,548

IBMCQ

5.90

1.10

5.80

6.00 1,200

10,054

85.00 IBMOQ 0.50

−0.40

0.50 0.55 5,373

14,901

IBMCR

2.27

0.56

2.25

2.30 6,516

14,198

90.00 IBMOR 1.85

−1.05

1.80 1.90 2,741

12,079

IBMCS

0.50

0.15

0.45

0.50 3,078

14,336

95.00 IBMOS

5.00

−1.00

5.00 5.10

426

4,213

IBMCT

0.05

−0.01

0.05

0.10

148

10,760 100.00 IBMOT 10.10

−1.49

9.50 9.70

62

1,913

IBMCA

0.05

0.00

0.05

0.15

2

4,150 105.00 IBMOA 15.30

−4.10 14.50 14.80

6

75

IBMCB

0.03

−0.02

0.05

0.05

16

3,473 110.00 IBMOB 18.30

−7.00

19.40 19.80

26

10

IBMCC

0.05

0.03

0.05

0.05

n.a.

410

115.00 IBMOC n.a.

n.a.

24.30 24.70

n.a.

n.a.

IBMCD

0.03

−0.04

n.a.

0.05

n.a.

81

120.00 IBMOD n.a.

n.a.

29.50 29.80

n.a.

n.a.

IBMCE

n.a.

n.a.

n.a.

0.05

n.a.

n.a.

125.00 IBMOE 34.00

0.00 34.40 34.70

1

n.a.

OPTIONS PRICE QUOTES

Apr 2009 Calls

Puts

Symbol Last Change Bid Ask Volume Interest Strike Symbol Last Change Bid Ask Volume Interest Trade Price Trade 11.8

0.90

11.60 11.80

458

IBMDQ 7.80 IBMDR 4.60

0.99

7.70 7.90

2,381

9,649

85.00 IBMPQ 2.40

0.73

4.50 4.70

1,672

11,644

90.00 IBMPR 4.20

2.40

0.60

2.30 2.40

3,405

21,552

95.00 IBMPS 6.90

IBMDT 1.05

0.30

0.95 1.05

992

IBMDP

IBMDS

3,221

80.00 IBMPP 1.30

15,196 100.00 IBMPT 12.01

−0.25 1.25 1.35

−0.23 2.40 2.45

−0.75 4.10 4.30 −1.20 6.80 7.00

−3.19 10.5010.70

1,527

12,260

1,060

13,098

2,145

10,644

254

14,501

1

2,553

Jul 2009 Calls

Puts

Symbol Last Change Bid Ask Volume Interest Strike Symbol Last Change Bid Ask Volume Interest Trade Price Trade IBMGP 14.23

0.83

14.50 14.70

IBMGQ 11.40

1.00

IBMGR 8.50

0.80

6.10

IBMGT 4.22

IBMGS

88

1,605

80.00 IBMSP 4.70

−0.26 4.60 4.70

58

3,457

11.20 11.40

317

8.40 8.50

600

1,883

85.00 IBMSQ 6.30

−0.60 6.30 6.40

264

4,300

5,110

90.00 IBMSR 8.67

−0.53 8.40 8.60

93

5,381

0.80

6.00 6.10

168

0.62

4.10 4.20

274

8,500

95.00 IBMSS 11.00

−0.70 11.0011.20

27

1,283

6,417

100.00 IBMST 15.31

−0.19 14.10 14.30

n.a.

848

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Oct 2009 Calls

Puts

Symbol Last Change Bid Ask Volume Interest Strike Symbol Last Change Bid Ask Volume Interest Trade Price Trade IBMJP 16.23

2.33

16.30 16.60

20

397

IBMJQ 13.40

2.70

IBMJR 10.59

2.29

80.00 IBMVP 7.00

−1.86 6.60 6.80

2

162

13.20 13.50

3

202

85.00 IBMVQ 9.30

−1.70 8.50 8.70

12

345

10.50 10.70

48

119

90.00 IBMVR 11.32

−0.18 10.70 10.90

n.a.

387

IBMJS

7.40

1.10

8.10 8.30

44

519

95.00 IBMVS 14.40

−0.60 13.3013.60

44

257

IBMJT

6.00

0.40

6.10 6.30

13

273

100.00 IBMVT 17.90

−2.30 16.3016.60

43

98

Jan 2010 Calls

Puts

Symbol Last Change Bid Ask Volume Interest Strike Symbol Last Change Bid Ask Volume Interest Trade Price Trade WIBAP 17.70

2.30

17.60 18.30

15

1,744

80.00 WIBMP 8.60

−1.80 8.40 8.70

n.a.

5.576

WIBAQ 14.10

2.30

14.60 15.20

1

1,151

85.00 WIBMQ 11.70

−0.90 10.40 10.70

8

2,616

WIBAR 11.63

1.23

12.20 12.50

n.a.

7,696

90.00 WIBMR 13.20

−1.60 12.7013.00

2

6,936

WIBAS

9.50

1.70

9.90 10.10

3

3,383

95.00 WIBMS 15.70

−0.90 15.3015.70

47

2,008

WIBAT

7.74

0.27

7.80 8.10

6

10,373 100.00 WIBMT 19.60

−3.70 18.3018.60

22

5,714 (continued)

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CHAPTER 14: OPTIONS MARKETS AND TRADING

Jan 2011 Calls Symbol

Last Change Bid Ask Volume Interest Strike Symbol Last Change Bid Ask Volume Interest Trade Price Trade

VIBAU

n.a.

n.a.

69.30 70.80

n.a.

n.a.

20.00 VIBMU 0.45

0.00

0.30 0.45

50

358

VIBAX

n.a.

n.a.

67.00 68.20

n.a.

n.a.

22.50 VIBMX 0.55

−0.25

0.45 0.60

100

204

VIBAE

n.a.

n.a.

64.50 65.80

n.a.

n.a.

25.00 VIBME 0.80

−0.12

0.65 0.75

2

105

VIBAV

n.a.

n.a.

59.30 60.80

n.a.

n.a.

30.00 VIBMV 1.20

−0.25

1.05 1.25

n.a.

883

VIBAG 49.50

1.00

54.70 55.90

n.a.

25

35.00 VIBMG 2.10

−0.08

1.65 1.90

20

163

VIBAW 47.70

1.90

50.10 51.30

1

41

40.00 VIBMW 2.95

0.05

2.40 2.60

1

236

−6.20 45.80 47.00

2

7

3.40

−0.10

3.20 3.60

n.a.

147

VIBAZ 39.20

−6.60

41.60 43.00

10

207

50.00 VIBMZ 4.70

−0.80

4.20 4.60

10

407

VIBAK 35.40

0.00

37.80 39.00

2

2

55.00 VIBMK 6.30

0.24

5.40 5.90

10

135

−0.10 34.30 35.40

3

92

60.00 VIBML 7.00

−0.80

6.80 7.20

1

323

VIBAM 28.20

−5.30

2

21

65.00 VIBMM 9.60

−0.70

8.30 8.70

44

293

VIBAN 24.80

−0.70 28.00 28.80

n.a.

254

70.00 VIBMN 11.60

0.00

10.00 10.60

25

331

VIBAP 20.20

−2.84 22.00 22.90

1.50

14.10 14.70

40

439

−0.96 16.30 16.90

117

748

18.80 19.40

n.a.

1,107

−1.90 21.50 22.20

11

636

VIBAI

VIBAL

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Puts

42.90

31.50

31.40 31.90

45.00 VIBMI

6

293

80.00 VIBMP 16.70

VIBAQ 18.50

0.96

19.90 20.30

50

479

85.00 VIBMQ 18.80

VIBAR 15.80

0.40

17.20 17.90

10

1,079

90.00 VIBMR 22.40

14.30

1.20

14.90 15.70

45

521

95.00 VIBMS 22.60

VIBAT 13.00

0.40

13.00 13.70

1

1,424

100.00 VIBMT 27.00

0.10

24.40 25.20

n.a.

676

VIBAB

9.80

0.30

9.50 10.10

4

812

110.00 VIBMB 35.50

0.10

30.70 31.60

24

512

VIBAD

6.60

−0.20

6.90 7.50

34

638

120.00 VIBMD 42.90

2.10

37.70 38.70

11

196

VIBAS

1.40

VIBAF

4.80

0.40

4.90 5.30

16

407

130.00 VIBMF 47.00

−4.00 45.40 46.20

3

193

VIBAH

3.60

0.60

3.20 3.70

1

599

140.00 VIBMH 54.80

−0.10 53.60 54.70

47

257

VIBAJ

2.25

0.05

2.25 2.50

67

672

150.00 VIBMJ 66.50

−2.00 62.30 62.90

42

227

Source: http://delayedquotes.cboe.com/options/options_chain.html?ASSET_CLASS=STO&ID_OSI=85502&ID_NOTATION=1551887

REGULATION AND MANIPULATION IN OPTIONS MARKETS

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14.7

Regulation and Manipulation in Options Markets

In the movie Mission: Impossible II (2000), a fictional character, Sean Ambrose, plans to release a virus and then profit by selling the antidote. “I want stock!” Sean thunders. “Stock options, to be precise.” And in the James Bond movie Casino Royale (2006), a fictional banker, Le Chiffre, provides a guerrilla group a safe haven for its funds. But Le Chiffre’s investments actually involve considerable risk. He buys puts on a successful company and then engineers a terrorist attack to sink the stock values. Stock options and market manipulation have entered pop culture! Options have long been associated with market manipulation. For example, during the 1920s, options gained notoriety when they were used in conjunction with stock price manipulation schemes. One abusive practice was for speculators with large holdings of the underlying stock to grant call options to stockbrokers (see Overby 2007). In return, brokers would recommend their clients purchase the stock. Another market manipulation scheme is buying out-of-the-money options and simultaneously buying the underlying shares to artificially increase the stock price so that the options end up in-the-money. Traders on the other side, who were getting ready to go out and celebrate happy hour on a late Friday afternoon, may suddenly find that their short out-of-the-money call options, which were seemingly worthless, suddenly end up in-the-money! We introduced the notion of a price bubble in Chapter 11, which happens when the stock price substantially deviates from its intrinsic or fundamental value. Bubbles typically involve a dramatic price rise followed by a drastic decline. Bubbles are associated with the dotcom boom of the 1990s and the recent real estate rise and collapse, and options are often connected with such episodes. A classic case was the tulip bulb bubble in seventeenth-century Holland. Dutch tulips are famous to this day, but their early history was fraught with speculation, greed, and fraud (see Malkiel 2003). The tulip bulb bubble story begins in 1593, when a botany professor brought tulip bulbs from Turkey to Holland, hoping to vend them at high prices. This market flourished, and tulips traded actively in the Netherlands in the years that followed. Some time later, many bulbs caught a benign virus that patterned their petals with bright stripes of contrasting colors. The public adored these “bizarres” and bid their prices up. Tulip bulb prices increased to a peak during the mid-1630s, when people from all walks of life, from noblemen to chimney sweeps, joined the fray. Some even bartered or pawned personal belongings to buy tulip bulbs with the hope of selling them at ever-increasing prices. Derivatives sprang up to feed the frenzy, and speculators used these derivatives (including options) for leverage—but they also helped the hedgers. Tulip growers and retailers bought calls and futures for protection against price increases. Such hedging strategies are practiced by many businesses even today. After months of outrageous prices and tumultuous trading, the market suddenly crashed in February 1637. Bad regulation and poor quality control have been blamed as the reasons for the crash. It’s no surprise that the regulators have historically been hostile to options. MacKenzie and Millo (2003) note that the proposal for the establishment of the

305

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CBOE was met with “instinctual hostility, based in part upon corporate memory of the role options had played in the malpractices of the 1920s.” CBOE’s first president, Sullivan, was told by one leading SEC official that he had “never seen a [market] manipulation” in which options were not involved. A former SEC chairman even compared options to “marijuana and thalidomide.” In a twist of fate, today’s regulators support options exchanges and consider them the lesser evil compared to the unregulated OTC derivatives markets.

14.8

Summary

1. Options trade because they provide payoffs different from forwards, futures, and the underlying stock. 2. Historically, options have been looked at with suspicion because of a cumbersome transaction process, small trading volume, high fees, counterparty risk, and the bad repute of market participants. The result was that investors surreptitiously traded a small number of contracts or shunned them altogether. The government’s response ranged from a grudging acceptance to an outright ban.

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3. Options markets got a boost in 1973 with the publication of the Black– Scholes– Merton model and the opening of the CBOE. Options trading has seen tremendous growth in the years that followed. This includes trading in options on interest rates, sector funds, exchange-traded funds, and indexes. 4. Equity options traded on the CBOE have the following features: a. Nature: American physical delivery options b. Contract size: one hundred shares c. Last exercise date: the third Friday of the expiration month d. Quotes: usually quoted in pennies ($0.01) for options prices below $3 and in nickels ($0.05) for higher options prices e. Strike prices intervals: $2.50 when the strike price is between $5 and $25, $5 between $25 and $200, and $10 for points over $200; strikes are adjusted for splits and recapitalizations but not for cash dividends (the CBOE has also introduced $1 strike price intervals and $2.50 strike price intervals, under certain terms and conditions, for many actively traded stocks) f. Strike prices: in-, at-, and out-of-the-money strike prices are initially listed; new series are generally added when the underlying trades through the highest or lowest strike price available g. Expiration months: two near-term months plus two additional months from the January, February, or March quarterly cycles h. Margin: short-term options buyers post the full price; long-term options buyers (maturing in more than nine months) can borrow up to 25 percent of the purchase price; options sellers are required to keep the full purchase price plus extra funds as cushion against risk

QUESTIONS AND PROBLEMS

14.9

Cases

Chicago Board Options Exchange (CBOE) (Harvard Business School Case

205073-PDF-ENG). The case discusses institutional details behind how options trade at the CBOE. International Securities Exchange: New Ground in Options Markets (Harvard

Business School Case 203063-PDF-ENG). This case examines the equity options market, the major parties involved, and the options trading process. Milk and Money (Kellogg School of Management Case, Case KEL343-PDF-ENG,

Harvard Business Publishing). This case considers how a family dairy firm can use regression analysis to choose the best hedges for its dairy products.

14.10 Questions and Problems 14.1. Briefly describe options trading that took place in 1500–1700 Amsterdam and

in nineteenth-century London and New York City. 14.2. a. Who was Russell Sage? b. Why was Sage called “The Old Straddle”? 14.3. Why did options trading fall into disrepute in the United States during the

early decades of the twentieth century? 14.4. During the nineteenth century, futures trading in the United States steadily

gained acceptance while options trading was associated with suspicion. Copyright © 2018. World Scientific Publishing Company Pte. Limited. All rights reserved.

a. Can you explain why this happened? b. How and when did this view of options trading get changed? 14.5. Briefly describe the events and developments that led to the founding of the

CBOE. 14.6. Describe two developments that took place in 1973 that made it a watershed

year in the history of options. 14.7. What is an option on a futures? Explain the workings of this derivative

contract. The next two questions are based on the following data from the CME Group’s website (prices as of 6:59:36 pm Central Standard Time on August 26, 2011). Spot price of gold is $1,830 per ounce. Contract size is one hundred ounces. Prices are per ounce. 14.8. a. Identify which calls are in-the-money, at-the-money, and out-of-the-

money. b. If you exercise a call with a strike price of $1,820, what is your payoff, and

what are your holdings of the futures contracts? c. For this call option on gold futures, what is the intrinsic value, and what

is the time value?

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Strike

Call

Put

1,820

85.20

107.80

1,825

83.20

110.90

1,830

81.30

113.90

1,835

79.40

217.10

1,840

77.50

120.20

14.9. a. There is an obvious mistake in the put price data—correct that first. b. Identify which puts are in-the-money, at-the-money, and out-of-the-

money. c. If you exercise a put with a strike price of 1,835, what is your payoff, and

what are your holdings of the futures contract? d. For this put option on gold futures, what is the intrinsic value, and what

is the time value? 14.10. Suppose Your Beloved Machines Inc. has a February cycle for options

trading.State the months for which regular equity options on YBM (which expire on the third Friday of the month) trade on the following dates: a. January 1

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b. January 27 c. March 1 14.11. State the dollar amount of margin you are required to keep with a broker

when trading one contract (on one hundred shares) of the following options on Your Beloved Machines Inc. YBM’s current stock price is $103. a. A long put worth $6 for an option maturing in six months b. A long call worth $8 for an option maturing in ten months 14.12. A long is generally associated with buying and a short with selling. Is it

counterintuitive that the put holder gets the right to sell? Explain your answer. 14.13. Your Beloved Machine’s current stock price is $90. YBM December 100 calls

trade for $6. a. Adjust the options prices and terms of the contract for a 4:1 split. b. Adjust the options prices and terms of the contract for a 3:2 split. 14.14. Are options on the CBOE adjusted for cash dividends or stock dividends or

both? 14.15. Use Figure 14.1. a. For the IBM April 2009 calls, is the call value increasing or decreasing in

the strike price?

QUESTIONS AND PROBLEMS

b. For the IBM April 2009 puts, is the put value increasing or decreasing in

the strike price? c. For the IBM March 2009 calls, which strikes are the most actively traded? d. For the IBM March 2009 puts, which strikes are the most actively traded? 14.16. What is exercise by exception with respect to CBOE options? 14.17. What is an options class? What is an options series? Give examples to illustrate

your answer. 14.18. What is open interest for a traded options contract? What does this tell you

about the trading interest in a particular options contract?

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14.19. Give two examples of market manipulation in the options market.

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15 Option Trading Strategies 15.1 Introduction 15.2 Traders in Options Markets 15.3 Profit Diagrams Copyright © 2018. World Scientific Publishing Company Pte. Limited. All rights reserved.

15.4 Options Strategies

15.6 Spread Strategies EXTENSION 15.1 Collars, Ratio Spreads, Butterflies, and Condors EXTENSION 15.2 Reinsurance Contracts and Call Spreads

Common Options Price Data

15.7 Combination Strategies

Naked Trades: The Basic Building Blocks

15.8 Summary

15.5 Hedged Strategies

15.9 Cases 15.10 Questions and Problems

TRADERS IN OPTIONS MARKETS

15.1

Introduction

While teaching options and futures for the first time, one of us (Chatterjea) was sharing some frustrations with a colleague, Mike Goldstein, about the difficulties in motivating students. Mike advised, “That’s not the way to teach derivatives. You should cover the interesting stuff, and show how to implement option trading strategies. Suppose that a takeover is going on, and explain how to set up a straddle to bet on the outcome.” He had hit the nail on the head! Focusing on equations like C(K1 ) – C(K2 ) ≤ K1 – K2 for K1 ≥ K2 is not the best way to attract interest. Moreover, this gives a false sense of mathematical sophistication while achieving little substance. It’s far more exciting to take options where the action is and discuss their role for speculating on takeover battles, placing leveraged bets on the markets, and protecting portfolios from market crashes. This is the approach we take here. This chapter studies some popular options trading strategies. We present profit diagrams for the basic options trades, hedged and spread strategies, including “butterflies” and “condors,” followed by strategies involving combinations of options. Along the way, we discuss how options relate to insurance contracts and present a bird’s-eye view of taxation and reporting requirements. After this chapter, you should be able to understand option related stories in The Wall Street Journal. The strategies we discuss are relevant to all players in the options market: the individual trader, the financial institution, the nonfinancial corporation, and even the government.

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15.2

Traders in Options Markets

Bill Lipschutz, who had taken one of the world’s first options courses from one of us (Jarrow) at Cornell University, joined Salomon Brothers Inc. in 1981, a prestigious Wall Street investment banking firm at the time.1 Though Salomon prided itself as one of the world’s most powerful trading companies, Lipschutz was disappointed to find options trading to be nonquantitative. He found that no one seemed to know the Black–Scholes–Merton model. Trading strategies were based on ad hoc criteria, for example, the head of the proprietary options trading desk said, “‘I went to buy a car this weekend and the Chevrolet showroom was packed. Let’s buy GM calls.’ That type of stuff.” Then, one day, a trader pulled him aside and gave away the great secret: “Look, I don’t know what Sidney is teaching you, but let me tell you everything you need to know about options. You like ’em, buy calls. You don’t like ’em, buy puts.” In the early 1980s, many Wall Street firms placed “leveraged outright bets” and prayed to the “Bitch-Goddess Success.”2 As the decade unfolded, however, most investment banks started assembling a team of derivative experts and running 1

This story is taken from the interview “Bill Lipschutz: The Sultan of Currencies” in Jack Schwager’s (1992) book The New Market Wizards: Conversations with America’s Top Traders. Lipschutz developed an early expertise in trading currency options, which began trading on the Philadelphia Stock Exchange in 1982. During his eight-year stint at Salomon, he routinely traded billions of dollars’ worth of currencies and became the “largest and most successful currency trader” in the firm. 2

Professor William James of Harvard University used the expression “Bitch-Goddess Success” in a letter to the British writer H. G. Wells.

311

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CHAPTER 15: OPTION TRADING STRATEGIES

proprietary trading desks to make profits. Goldman Sachs and Company, a key player in this transformation, lured Professor Fischer Black from the Massachusetts Institute of Technology and made him a partner. Black helped usher in the migration of many finance professors and numerous talented PhDs in the natural and engineering sciences into Wall Street firms. The most mathematical of these recruits came to be variously viewed as “quants” and “rocket scientists” whose job was to apply their math mastery to make profits. Salomon did not lag behind. It pioneered the mortgagebacked securities market during the 1970s, arranged the world’s first swap contract in 1981 (see Chapter 22), and built an active derivatives shop. Its consultants included Nobel laureate Robert Engle, who was hired to develop volatility forecasts and devise trading strategies based on cutting-edge econometrics. Much of our discussion of options strategies in this chapter will be from the perspective of an individual investor. Financial institutions also trade options to bet on the markets and capture arbitrage profits. Hedge funds and proprietary trading desks at investment banks also implement the strategies we describe. Warren Buffett noted in his 2008 chairman’s letter in the annual report of Berkshire Hathaway Inc. that his company had entered into 251 derivatives contracts, including writing fifteen-and twenty-year maturity European put options on four major indexes (the Standard and Poor’s [S&P] 500, London’s FTSE 100, Europe’s Euro Stoxx 50, and Tokyo’s Nikkei 225). He believed that each of these contracts was “mispriced at inception [emphasis added], sometimes dramatically so.” Doesn’t this smack of arbitrage? Affirming his belief that “the CEO of any large financial organization must be the Chief Risk Officer as well,” Buffett declared that he had initiated these positions, would continue to monitor them, and would take full responsibility if the company lost profits on these derivatives trades. We discuss later in the chapter how traders (including insurance companies) use options to hedge and speculate on catastrophic risks. Many foundations and endowments, mutual funds, and pension funds actively trade options. Options often underlie financial engineering techniques used by nonfinancial companies, which buy inputs to produce outputs. Sometimes governments use options. Recall the example at the beginning of Chapter 14, in which Lawrence Summers and Zhu Rhongji discussed the feasibility of using put options to defend the value of the Chinese currency and how the US government used warrants to design a highly effective financial rescue package for troubled banks. Options trading strategies adopted by different kinds of traders get regularly mentioned in the financial press and addressed in case studies.

15.3

Profit Diagrams

As pictures say a thousand words, we use profit diagrams as our primary tool for discussing options trading strategies. Recall that profit diagrams place the stock price on the expiration date S(T) on the horizontal (x) axis and the net profits from trades on the vertical (y) axis. The profit or loss is computed by vertically adding up the profits and losses for each security within the trade. Wherever convenient, we show the strike price K, the maximum profit (or loss), and the stock price corresponding

OPTIONS STRATEGIES

to the breakeven point (BEP; or the zero-profit point), where the trader recoups the option premium. Absent market frictions, the zero-sum nature of options trading ensures that the BEP is the same for both the buyer and the writer. Profit diagrams have some limitations. First, you can only draw them for European options. American options have the risk that one or more legs of the strategy may get exercised early and disappear from the diagram. Second, they do not apply before expiration because the time value of the options (see Chapter 5) moves their profits away from those on the expiration date. Third, we must have just one stock as the underlying. You cannot draw profit diagrams for a portfolio of options on different stocks. Still these diagrams are quite useful. If American options remain unexercised until expiration, they have the same payoffs as European options. Although most exchangetraded options in the US are American options, early exercise isn’t very common. The next chapter will demonstrate that options are usually worth more alive (i.e., unexercised) than dead (exercised). Profit diagrams provide a useful snapshot of the possibilities on the option’s maturity date. We focus on equity options. With minor modifications, you can similarly analyze other options, including those on commodities, indexes, and even futures and forwards. For now, we examine simple options trading strategies.

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15.4

Options Strategies

A story titled “Options Trading Grew Up in 2009” (Wall Street Journal, January 4, 2010) reported that many new options traders entered the market in 2009. What attracted these options traders? The article identified several factors. Many investors traded options to better manage their portfolio’s risks after suffering losses. Some adopted complex strategies that “helped them to reduce trading costs and limit the impact of ‘decay,’ which describes the pace at which options lose value as they approach expiration.” To prevent losses, they bought put options or “put spreads,” and to generate income, they tended to sell “covered calls.” This and the next two sections discuss these options trading strategies.

Common Options Price Data Most of our options trading strategy examples will use the following common options pricing data (COP data) for a fictitiously named company, OPSY, which is the Dutch name for options: ■

The current stock price S is $22.50.

■

The time to maturity T is six months.

■

The strike prices K are $17.50, $20, $22.50, and $25.

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TABLE 15.1: European Option Price Data for OPSY Strike Price

Call Price

Put Price

K0 = $17.50

$5.50

$0.10

K1 = 20.00

3.50

0.50

K2 = 22.50

2.00

1.50

K3 = 25.00

1.00

3.00

Stock price S = $22.50; time to maturity T = 6 months; risk-free interest rate r = 5 percent per year.

■

The continuously compounded, risk-free interest rate r is 5 percent per year.

■

The European options prices are given in Table 15.1. These assume a volatility equal to 0.26.

We ignore market imperfections such as transaction costs and also interest earned on the option’s premium. Usually these are small. Of course, you can easily add them to the profit or loss amounts.

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Naked Trades: The Basic Building Blocks The basic option trading strategy building blocks are stand-alone positions containing a single option. Traders refer to these as naked trading strategies (who said options are boring?). Long naked trades are outright purchases because you are not holding an offsetting position in the underlying stock. Short naked trades are also known as uncovered strategies because you are not covering (by reducing the risk of) the trade by simultaneously taking a position in the underlying stock.

EXAMPLE 15.1: Going Long and Selling Short: Stocks and Options Before illustrating the four uncovered positions for European call and put options (with strike prices of $22.50), we draw profit diagrams for long and short stock trades (see Figure 15.1). You can evaluate option trades vis-á-vis these key references. Naked option trading strategies are popular among traders.

Long and Short Stock ■

As stocks are limited liability assets, the maximum loss is the price paid. Long OPSY’s profit diagram is a straight line that emanates from a loss of $22.50 along the vertical axis. It rises at a 45 degree angle, cuts the horizontal axis at the breakeven point of $22.50, and continues upward with infinite profit potential.

■

The short stock’s profit diagram is the mirror image of the long stock across the x axis. An oft-quoted Wall Street saying attributed to the notorious financier Daniel Drew explains the riskiness of short selling: “He who sells what isn’t his’n, buys it back or goes to prison.”

OPTIONS STRATEGIES

Long Call ■

Buying OPSY 22.50 European calls for $2 gives the right to buy the stock for $22.50 at time T. Recall from Chapter 5 that the profit diagram is a horizontal line starting from (a maximum loss of) $2 on the vertical axis. Then, at the strike price of K = $22.50, it shifts upward at a 45 degree angle, breaks even where the stock price S(T) equals the strike price plus call premium (BEP = $24.50), and increases with an unbounded profit potential.

FIGURE 15.1: Profit Diagrams for the Basic Security Blocks (Naked Strategies) Profit

Profit Long stock 1

22.5

1 22.5

0

45°

22.5

0 S(T) Stock price at expiration

–22.5

S(T) 1 1

Short stock

Call and put strike price K = $22.50 Profit

Profit

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Long call

21

Long put 22.5

24.5 2 0 –2

1.5

45° S(T) 22.5 Short call Breakeven point for call = $24.50

■

–1.5 –21

45°

S(T)

21 Short put Breakeven point for put = $21

Applications and uses: - Buy calls to benefit if the stock goes up while limiting losses to the premium if the stock goes down. You can buy OPSY stocks for $22.50 per share. Alternatively, you can buy OPSY 22.50 six-month calls for $2 per share. Each trade has its pros and cons. Action on the call happens during its short life span. By contrast, you can hold on to the stock forever in the hope of a price increase. If OPSY goes down to $15, you lose $7.50 per share but just $2 on the call. If OPSY goes up, you immediately start benefiting on the long stock, but you have to wait until the stock price crosses the breakeven point of $24.50 before profiting on the call.

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- Let’s look at percentage returns. A $7.50 decline is a 33.33 percent loss on the stock investment, but a $2 loss on the premium is a 100 percent loss on the option. If the stock goes up to $30, then there is a 33.33 percent profit on the stock but a (30 – 24.50)/2 = 275 percent profit on the call. The dollar gain or loss is higher for the stock, but the percentage gain or loss is higher for the option. This happens because options are leveraged investments. - Buy calls to speculate on events, corporate developments, and sector and economic performance.

Short Call ■

Selling a call option is also known as shorting or writing a call. A short call’s profit diagram is the mirror image across the x axis of that for a long call.

■

Naked call writing is one of the riskiest strategies, and only seasoned option traders with sufficient financial resources are allowed to employ this strategy.

Long Put ■

A put buyer loses the entire premium of $1.50 if the stock price stays above the strike but starts recouping his investment when S(T) dips below this mark. The breakeven point is at $21, where the $1.50 gain exactly offsets the premium paid. Profit is maximized when S(T) hits zero. Here a worthless stock is sold for $22.50, and deducting the premium arrives at a profit of $21. - Put purchasers bet on the downside but limit their losses in case the bet goes wrong. As with calls, a long put has similar pluses and minuses vis-à-vis a short stock trade.

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Short Put The uncovered put writer has the obligation to buy OPSY for $22.50 on assignment of an exercise notice. A put writer is uncovered if he does not have a short stock position or has not deposited cash equal to the strike to form a cash-secured put. Naked put sellers take losses during market crashes. One of the most notorious examples happened during the Black Monday crash of October 19, 1987. Buoyed by the large stock market gains during the mid-1980s, naked put sellers viewed put premiums as free cash. But when the Dow fell 22.61 percent on this fateful day, many of these put sellers were wiped out. ■

Naked put writers often have two objectives: to receive premium income and to acquire stock at a cost below its market price at some future date.

■

Short put strategies are bullish.

To summarize, naked options trades generate profits that are similar to those created by trading stocks without their downsides. But they are created for a limited time and require regular payment of premiums to maintain the position over long time periods.

HEDGED STRATEGIES

15.5

Hedged Strategies

Hedged strategies combine options and stocks in ways that reduce the overall risk of naked option trades. They are usually covered strategies because they hedge option transactions with the stocks that protect (cover) them.

EXAMPLE 15.2: Hedged and Covered Strategies

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Covered Call Writing (Long Stock plus Short Call) ■

Covered call writing combines a short call with the underlying stock. Figure 15.2 illustrates this 1:1 hedge strategy. - At the origin, the long OPSY trade incurs the maximum loss ($22.50), but the short call retains the entire call premium of $2. The combined profit is – 22.50 + 2 = – $20.50. - As the stock price increases, the short call’s profit graph (a dotted line parallel to the horizontal axis) remains constant at $2, but the long stock’s profit graph (an upward sloping dashed line) rises by a dollar for each dollar increase in S(T). The resulting profit diagram (shown by a solid line) rises at a 45 degree angle, until the point at $22.50. - At the strike price of $22.50, the profit is $2 (from short call) + 0 (from long stock) = $2. For higher values of S(T), the short call’s profit decreases at a negative 45 degree angle, which neutralizes the long stock position’s increase. Consequently, the covered call’s profit diagram is a flat line for all higher values of OPSY.

■

Doesn’t the profit diagram for this concoction look like the diagram for a short put position? The next chapter will explain why this happens.

■

The timing of the trades gives this strategy different names. It’s called a buy–write if one simultaneously buys the stock and sells the call, but it’s an overwrite if one sells the call after purchasing the share.

■

Buy–write’s popularity prompted the CBOE to develop an innovative benchmark for measuring the performance of such strategies. Released in April 2002, the award-winning CBOE S&P 500 BuyWrite Index (BXM) is based on the total returns from hypothetically holding the S&P 500 stock index portfolio and writing a slightly out-of-the-money, one-month maturity call option on the S&P 500 index.

■

Traders who write covered calls generally have two objectives: collecting a premium or hedging a stock trade. By accepting the premium, the writer surrenders the opportunity to benefit from a stock price rise above the exercise price. The call premium gives the writer some cushion against a stock price fall, but he remains vulnerable to losses from a deep decline in the stock. Covered call writing works well in neutral markets, particularly when the volatility is high, which translates into higher call prices.

■

Covered call writing is a more conservative strategy than an outright stock purchase. A director of investments at a pension fund went as far as to comment that covered call writing is one of the most conservative strategies possible.

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FIGURE 15.2: 1:1 Hedges: Covered Call Writing (Buywrite or Overwrite) and Covered Put Buying (Married Put or Protective Put) Profit

Long stock 20.5

Covered call writing

2 0

S(T) 22.5 Short call 1 1 –20.5 –22.5 Profit 21

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Long put Covered put buying 22.5 0 –1.5

S(T) 24 1 1 Long stock

–22.5

■

Even the government views covered call writing as a less risky strategy. You can write covered calls in some tax-advantaged retirement savings accounts. In June 2009, the Securities and Exchange Commission approved allowing employees to sell calls against unexercised holdings of employee

HEDGED STRATEGIES

stock options (which are call options on the stock that are granted by the company as a form of noncash compensation; see “Stock Options Opened for ‘Call Writing,’” Wall Street Journal, June 26, 2009).

Covered Put Buying (Long Stock plus Long Put) ■

Another highly popular strategy, covered put buying has long been used to justify the trading of options: to buy insurance by hedging a long stock with an option. Figure 15.2 illustrates this strategy.

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- At the origin, the long OPSY trade loses $22.50 but the long put position has a gain of $21, giving a profit of – $1.50, which is the maximum loss. - For each dollar increase in S(T), the long put decreases the covered put strategy’s profit by a dollar, giving a profit parallel to the horizontal axis. - When S(T) reaches K = $22.50, the long put’s profit changes and becomes parallel to the x axis, but the long stock position keeps increasing. As a result, the profit rises at a 45 degree angle, signifying boundless profit potential. ■

As with the previous strategy, the timing of the trades gives the strategy different names. It is called a married put (from an old Internal Revenue Service ruling, states the Options Industry Council website) if one simultaneously buys the put and the stock, but a protective put if one buys the put to protect the downside risk of a stock previously purchased.

■

Again, the profit diagram looks like another option trade, a long call position. We will explain this in the context of put–call parity in the next chapter.

■

Traders employ covered put strategies for several reasons: - Establishing a minimum selling price for the stock. No matter how far the stock declines, your loss is limited to the put premium plus the difference, if any, between the stock’s purchase price and the put’s exercise price. In this case, the minimum selling price is (Strike price – Put premium) = 22.50 – 1.50 = $21. - Buy puts to protect unrealized profit in a long stock position. Suppose you bought OPSY when it was trading for $15 and it has now reached $22.50. You can set up a protective put position to hedge your profit against short-term price declines. No matter how far the stock falls, you can always exercise the put and sell the stock at the strike price $22.50. Your profit never falls below [(Strike price – Stock purchase price) – Put premium] = (22.50 – 15) – 1.50 = $6.

Covered Call Buying (Short Stock plus Long Call) Suppose you are bearish and sell OPSY short. This is a high-risk strategy. What’s your protection if the stock rallies instead? One can hedge the risk by buying a call (perhaps an out-of-the-money call like OPSY 25, which has a price of $1).

Covered Put Writing (Short Stock plus Short Put) A less popular strategy, covered put writing tries to generate some income for the short sellers.

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The 1:2 Hedge The previous four strategies considered 1:1 hedges (see Figure 15.2), in which one option was covered by one stock. You can also create hedged strategies by combining unequal numbers of options. A 1:2 hedge combines a long stock with two short calls (see Figure 15.3) or two long puts. The profit diagram looks similar to the diagram of a straddle discussed later in this chapter. These strategies have long been used by dealers who write options; for example, when options are at-the-money, a call writer may buy half a share for each call sold, whereas a put writer may sell half a share for each put sold. The precise quantity of stocks to hedge options trades to remove all of the portfolio’s price risk is given by the stock’s delta, which is discussed in Chapters 17, 18, and 19 and in considerable detail in Chapter 20.

FIGURE 15.3: Long Stock and Two Short Calls (1:2 Hedge) Profit Long stock 4 0

26.5

18.5

S(T)

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22.5 1:2 Hedge

–18.5

Two short calls

–22.5

15.6

Spread Strategies

Oxford University professor John L. Austin asserted during a lecture that although there are many languages in which a double negative makes a positive, no language uses two positives to express a negative statement. Seated in the audience was another eminent philosopher, Columbia University professor Sidney Morgenbesser. He dismissively dissented, “Yeah, yeah,” and the tone of his voice disproved the claim. In finance, we can use a variation of this venerable philosopher’s assertions to describe a spread strategy in which two risky trades combine to form a less risky portfolio. This happens because a spread strategy subtracts “similar” risks, making the end product safer than either of the two originals. Because spreads are two-sided hedges

SPREAD STRATEGIES

that combine two options of the same type but on opposite sides of the market, the exchanges allow spreads to have lower margins than naked trades. A spread gives a smaller profit if the underlying moves in one direction but a tiny loss otherwise. Choice of which option to buy and which to sell determines whether it is a bullish or a bearish spread. Spreads come in three basic types: (1) vertical spreads (also called money, perpendicular, or price spreads) involving options that have different strike prices but expire on the same date; (2) horizontal spreads (or time or calendar spreads) involving options that have the same strike but different maturity dates; and (3) diagonal spreads involving options that differ both in terms of strike price and maturity date. Different maturity dates prevent representing diagonal and horizontal spreads in standard profit diagrams. Instead, traders compute options sensitivities to understand the behavior of such portfolios. Chapter 20 explains these tools.

EXAMPLE 15.3: Bull and Bear Spreads

Bull Spread

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■

You can create a bullish vertical spread by buying an option and simultaneously selling an option of the same type but with a higher strike price, where both options have the same underlying security and the same date of maturity.4 Traders set up bull spreads when they are optimistic about the underlying stock. We create a bull call spread by trading OPSY calls from our COP data, as follows: - Buy the OPSY call with a strike price K1 = $20 for $3.50 and sell a call with K2 = $22.50 for $2 and draw their profit graphs (see Figure 15.4). - If the stock price at expiration is zero, the loss of $3.50 on the first call is partially offset by a $2 gain on the second call, giving a loss of $1.50. - The loss stays constant until S(T) reaches $20. Beyond this, the long call’s profit graph increases at a 45 degree angle, while the short call’s profit graph remains flat. Consequently, the profit graph for the bull spread, which was $1.50 below the horizontal axis, moves up dollar for dollar for each dollar increase in S(T), cuts the x axis at the BEP = (20 + 1.50) = $21.50, and reaches $1 when S(T) reaches $22.50. - Beyond $22.50, the short call position decreases at a negative 45 degree angle. This decline neutralizes the upward rise from the long call, and the net result is a flat line that is $1 above the x axis.

■

This is a debit spread because it costs money to create the spread, as opposed to in a credit spread, where you receive money in the creation process. You create a credit spread when you enter into a bullish put spread by buying a put and simultaneously selling another put with a higher strike price. For example, by buying the OPSY 20 put and selling the OPSY 22.50 put, you create a bull spread that has a maximum loss of $1.50, a BEP at $21.50, and a maximum profit of $1—a profit graph that happens to be identical to the graph of our previous bull call spread (see Figure 15.4).

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FIGURE 15.4: Vertical Bull Spread with Calls Profit

20

2 1 0 –1.5 –3.5

Long call (K1 = 20)

21.5

Bullish call spread S(T) 22.5

Short call (K2 = 22.5)

FIGURE 15.5: Vertical Bear Spread with Puts Profit

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21

Long put (K2 = 22.5) 1.5 0.5 0 –1 –1.5

Bearish put spread

21.5

S(T) 22.5 20

–19.5

Short put (K1 = 20)

Bear Spread ■

A bearish vertical spread is created by buying an option and simultaneously selling an option of the same type, but with a lower strike price, where both options have the same underlying security and maturity date. We create a bear put spread by trading OPSY puts as follows: - Sell an OPSY put with strike price K1 = $20 for $0.50 and buy a put with K2 = $22.50 for $1.50 and draw their profit diagram (see Figure 15.5).

SPREAD STRATEGIES

323

- At the origin, the long OPSY put trade gains $21, but the short put position loses $19.50, giving a payoff of $1.50, which is the maximum profit. For higher values of S(T), the profit is a flat line because the long put declines but the short put rises, and the two effects cancel each other. - The profit stays constant until S(T) reaches $20. Beyond this point, the long put’s profit continues to decline at a 45 degree angle, whereas that for the short put remains flat. The profit graph for the bear spread, which was $1.50 above the horizontal axis, decreases dollar for dollar for each dollar increase in S(T), cuts the x axis at the BEP (20 + 1.50) = $21.50, continues declining, and reaches –$1 when S(T) touches $22.50. Beyond this, the long put position also becomes a flat line, and the payoff becomes a line parallel to the x axis. - Notice that the bear put spread of Figure 15.5 is a mirror image of the bull put spread of Figure 15.4 (which is the same as a bullish call spread) across the horizontal axis. This is no surprise because the two spreads involve identical options that are traded in opposite directions. - You can similarly create a bear call spread by reversing the trades that led to the creation of the bull call spread. You can create profit diagrams that look like vertical spreads by trading the underlying asset and options. This is demonstrated in Extension 15.1.

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4 For a bullish horizontal spread, buy an option and sell another option of the same type and strike but with shorter life. For a bullish diagonal call spread, buy an option with a lower strike price and longer time to maturity than the one that is sold. Do the opposite for bearish spreads.

A spread can be used as a starting point for understanding more complex strategies. For example, you can begin with a vertical spread and tinker with it by adding more options of the same type. This leads to ratio spreads, butterfly spreads, condors, and other options trading strategies, some of which are popular and others obscure. Extension 15.1 shows how to do this. Moreover, one can find options spreads embedded in other contexts. For example, a reinsurance contract is equivalent to a call spread. This connection is explored in Extension 15.2.

EXTENSION 15.1: Collars, Ratio Spreads, Butterflies, and Condors Consider a simple example. Suppose the portfolio your boss manages tries to mimic the S&P 500 index, but it allows some tinkering. You advise buying some puts on the index with strike K1 with cost $p per option and financing this by selling calls with the same price $c but a higher strike price K2 . Ext. 15.1 Fig. 1 shows that the profit graph of this zero-cost collar (long stock, long put[s], and short call[s], where the total cost of the calls equals that of the puts) is a spread. The strategy has its trade-off. You are pleased if the underlying declines and the put protection works but terribly glum if the underlying rises and the gains are surrendered to the call buyer. Collars are similar to zerocost collars but do not require the cash flows from the calls and puts to negate each other. You can establish a collar on a long stock position by adding a covered call and a protective put.

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You can create butterfly spreads and condors by modifying a vertical spread. We demonstrate this by first drawing a vertical bull spread with our COP data. Then we create a whole range of spread strategies by adding options of the same type (calls with calls or puts with puts), but with different strike prices.

EXT. 15.1 FIG. 1: A Zero-Cost Collar Profit

Long stock

Long put K1

Zero-cost collar

c 0 –p

S(T) K2

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Short call

Here call price c = put price p

EXT. 15.1 EX. 1: Examples of a Call Spread, a Ratio Spread, Butterflies, and Condors

Bull Call Spread One can create a bull call spread similar to the one in Example 15.3 by buying one OPSY 17.50 call for $5.50 and selling one OPSY 20 call for $3.50. The profit diagram is initially a flat line with a loss of $2, rises at a 45 degree angle as the stock price goes from $17.50 to $20, breaks even at $19.50, and again becomes a flat line at a profit of $0.50 when S(T) exceeds $20 (see Ext. 15.1 Fig. 2).

Ratio Spread ■

Now sell another OPSY call with strike price $20. By selling two calls, you collect a premium of $3.50 × 2 = $7, and the profit diagram declines by $2 for each dollar increase in S(T) beyond $20. This creates a ratio spread, which is defined as an options trading strategy in which the number of calls (or puts) purchased is

SPREAD STRATEGIES

325

different from the number of calls (or puts) sold. The profit graph for this one-by-two call spread is a flat line emanating from (– 5.50 + 7) = $1.50 on the vertical axis, which then rises at 45 degrees when S(T) exceeds $17.50 and attains the peak value of (1.50 + 2.50) = $4 when S(T) is $20 (see Ext. 15.1 Fig. 2).5 Beyond this, the long call moves the graph up, but the two short calls drag it down twice as fast, resulting in a line that declines at a negative 45 degree angle.

Butterfly Spread ■

A butterfly spread is created by trading four options of the same type (all calls or all puts) with three different strike prices: two options with extreme strike prices are bought (written) and two options are written (bought) with the middle strike price (see Ext. 15.1 Fig. 2).

■

A butterfly spread is a two-sided hedge that bets on the volatility of the underlying security: a long butterfly spread makes a small profit if the volatility is low and a small loss otherwise. The profit graph for this long butterfly call spread starts at a maximum loss of (–5.50 + 7 – 2) = – $0.50 along the vertical axis. Initially a line parallel to the horizontal axis, the profit graph starts increasing when S(T) exceeds $17.50, breaks even at $18, attains a maximum value of $2 when S(T) is $20, declines and cuts the x axis again at $22, falls to – $0.50– $0.50 at $22.50, and again becomes a line parallel to the x axis for all higher values of S(T).

■

A butterfly spread is so named because the profit diagram looks like a butterfly’s wings. If you reverse the preceding trades, you will create a short butterfly spread. You can also create butterfly spreads with puts.

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Condor ■

Thinking that OPSY might fluctuate more than anticipated, you decide to expand the region over which you want to profit. You modify the butterfly spread by (1) keeping a long call with K0 = $17.50 as before, (2) (instead of two short calls at the same strike) selling one call at the next higher strike price K1 = $20 and another at K2 = $22.50, and finally, (3) buying a call with the uppermost strike K3 = $25 (see Ext. 15.1 Fig. 2). This creates a condor spread, which has four options with four strike prices: long two options with extreme strikes and short two options with strike prices in the middle, and vice versa. The wings of the condor, the largest flying land bird in the western hemisphere, inspire the name.

■

Our condor’s profit diagram begins on the y axis at (–5.50 + 3.50 + 2 – 1) = – $1, the amount obtained by adding up the premiums. Initially, it’s a line parallel to the x axis because the profits from the calls are flat at this point. When S(T) goes beyond 17.50, the call with the lowest strike becomes active and increases the payoff. The profit graph reaches a maximum value of $1.50 when S(T) is 20. It becomes a flat line for S(T) lying between $20 and $22.50 because the call with a strike of K0 = $17.5 pulls it up but the call with K1 = $20 pushes it down. Beyond $22.50, the third call (with K2 = $22.50) kicks in, and the profit graph falls. Finally, when S(T) exceeds $25, the fourth option (with K3 = $25) also becomes active and increases the payoff: with two calls pulling it up and two pushing it down, it again becomes a flat line. You can also create a condor with puts.

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EXT. 15.1 FIG. 2: Bull Call Spread, Ratio Spread, Butterfly, and Condor Profit Long call (K0 = 17.5)

17.5 3.5 0.5 0 –2 –5.5

Bullish call spread S(T) Short call (K1 = 20) 20

Profit 7 4 1.5 0 –5.5

Long call (K0 = 17.5)

17.5

S(T)

1 20

1 Call ratio spread

2 1

Two short calls (K1 = 20)

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Profit Long call (K0 = 17.5) Long call (K2 = 22.5)

7 17.5 2 0 –0.5 –2 –5.5

22.5

20

S(T) Butterfly call spread

Two short calls (K1 = 20) Profit

17.5

25

1.5 0 –1

S(T) Condor spread 20 22.5

SPREAD STRATEGIES

327

5 A call ratio spread typically involves buying calls and selling a greater number of calls at a higher strike price. If you sell calls and buy a greater number of calls at a higher strike price, then you create a call backspread.

EXTENSION 15.2: Reinsurance Contracts and Call Spreads Reinsurance contracts on a loss causing event (e. g. hurricanes) are equivalent to a call spread in which the call options are written on the losses of the insured event. To understand this equivalence, consider the following example.

Risks Facing the Insurance Industry

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Consider some of the risks in the insurance industry. ■

Dearth of natural hedges. Insurance is a high risk business that cannot be hedged using traded financial securities. Insurance companies employ applied mathematics and statisticians called actuaries, who calculate the likelihood of loss events and fair premiums for insuring these risks.

■

Diversification and the law of averages. Diversification is the industry’s key mantra. Insurers try to manage risks by diversifying their losses across a large number of policyholders. The business survives on the law of large numbers, which says that as more and more independent and identically distributed observations are collected, the average gets closer and closer to the mean value for the population. Although no one can foretell whether any particular insured entity will suffer a loss, actuaries can estimate with reasonable accuracy the expected losses in a large population. They compute fair premiums based on these expected losses. Insurance companies lose money when the realized losses exceed the expected losses.

■

Reinsurance. For catastrophic events like major earthquakes and hurricanes, the law of large numbers does not apply because these events are not independent and identically distributed across the population. Insurance companies hedge their losses on such catastrophies by buying their own insurance. This is called reinsurance. We discuss reinsurance after the following example.

EXT. 15.2 EX. 1: Hedging Hurricane Risk ■

Catastrophic Insurance Corporation (CatIns Corp., a fictitious name) sells homeowner’s insurance contracts to one hundred thousand homes along the shores of the Gulf of Mexico in the states of Florida, Alabama, Mississippi, Louisiana, and Texas. The contract provides protection against hurricane risk, which is the risk of losses coming from powerful storms, and it has the following features: - Annual premium $15,000. - Fixed-amount deductible $10,000. - Maximum coverage amount $300,000 (which is less than the value of each home). - Contract pays for losses from one hurricane in a given year. For simplicity, assume that if a hurricane hits, it does the same dollar damage to all homes covered by the insurance policy. These losses manifest the

328

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inability of the insurance company to diversify these losses in a large pool of the insured. Also assume that underwriting costs and investment income net out to zero. ■

We draw the profits for CatIns in a modified profit diagram in which “profit per contract” for the insurance company is plotted (as before) along the vertical axis and the underlying “loss per home” (instead of underlying asset price) is depicted along the horizontal axis. Of course, you can draw the profits in a usual profit diagram. However, this representation is more natural in the context of insurance, where losses are key. We determine the profits as follows (see Ext. 15.2 Fig. 1; both variables are graphed in thousands of dollars).

EXT. 15.2 FIG. 1: Call Spread Diagram Profit per contract (thousands of dollars) 15 12 0

25 10

–78

100 22

300 Loss per home (thousands of dollars) CatIns’s profits (with reinsurance)

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CatIns’s profits (no reinsurance) –275

No Hurricane CatIns keeps the entire premium of $15,000, which is shown as 15 in the diagram. For one hundred thousand homes, this turns out to be $1.5 billion.

If Hurricane Damage Happens For the first $10,000 in losses, the company pays nothing. The profits (shown by a shaded line) remains parallel to the horizontal axis at $15,000. For higher losses, the company pays a dollar for each dollar increase in loss. The profit line declines at a 45 degree angle and cuts the x axis when the total loss reaches $25,000, shown as 25 in the diagram. Beyond this, the profit declines until the total loss on each home reaches the maximum loss of $300,000, where CatIns pays $275,000. This is a profit of $27.5 billion, a staggering sum considering that the company earned only $1.5 billion in premiums.

Insurer Buys Reinsurance CatIns can buy reinsurance that pays for losses to a home over $100,000 for a premium of $3,000 payable to a reinsurance company. Then, the new payoff line emanates from the y axis at (15,000 – 3,000) = $12,000, stays

COMBINATION STRATEGIES

329

parallel to the x axis until “loss per home” reaches $10,000, goes down thereafter at a 45 degree angle, cuts the horizontal axis at $22,000, reaches a maximum loss of (100,000 – 22,000) = $78,000 when “loss per home” reaches $100,000, and again becomes parallel to the x axis because the reinsurance company pays for the higher losses.

Equivalence to a Call Spread This loss diagram shows that the profits to CatIns with reinsurance are equivalent to a call spread on the hurricane losses. The company makes a limited profit if the hurricane’s losses are mild, and the company incurs losses that are capped at $7.8 billion when large hurricanes hit homeowners.

Reinsurance Markets

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Reinsurance has a long history. One of the earliest known reinsurance contract agreements was written in Latin and signed by Italian merchants on July 12, 1370. Reinsurance helps insurance companies manage tail risk, which refers to the risk of occurrence of infrequent events that cause large losses when they occur. Tail risk is so named because these large loss events lie on the extreme end (“tail end”) of a probability distribution of such loss events. As illustrated by the example, an insurance company’s profit diagram under a typical reinsurance contract is equivalent to a call option spread. The call options are written with a one-year maturity on the aggregate losses realized by the insurance company, where the strikes correspond to the initial deductible and the payment cap. Given the call spread analogy, reinsurance contracts can be priced using the option pricing methodologies presented in Chapters 17–20.

15.7

Combination Strategies

“A double privilege pays a profit, no matter which way the market goes, and costs $212.50,” declared an advertisement in 1875 in the book Secrets of Success in Wall Street by Tumbridge and Co., Bankers and Brokers, which had its office on 2 Wall Street, New York City at that time. An ancestor to today’s booklets that aim at informing and attracting traders to options, this forty-eight-page volume describes the workings of the New York Stock Exchange and the over-the-counter options market. The traded contracts went by the name stock privileges and were essentially American options: calls, puts, spreads, and straddles that were customarily written on one hundred shares of stock and matured in thirty days. Tumbridge charged $100 as the premium for a call or a put and a $6.25 broker’s commissions for each leg of the trade. Double privilege referred to spreads and straddles, which were created by trading two options, and hence charged two commissions. Straddles and strangles are examples of combination strategies that combine options of different types on the same underlying stock and expiring on the same date, where the options are either both purchased or both written. For example, buying a call and a put with the same strike price creates a straddle (called a putto-call strategy in London); if their strike prices are different, you get a strangle.

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Straddles and strangles are volatility plays. Options traders use them to bet on risky future events affecting a company that can impact the stock price in either a positive or negative direction. Traders sometimes write them when they expect neutral markets.

EXAMPLE 15.4: Straddles and Strangles

Straddle ■

You buy a straddle (a long straddle or a bottom straddle) by buying a call and a put as follows: - Buy OPSY 22.50 options, the call for $2 and the put for $1.50, and draw the profit diagram (see Figure 15.6).

FIGURE 15.6: Straddles and Strangles—Examples of Combination Strategies Profit 21 19

Long put Long call Straddle

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22.5 19

0 –1.5 –2 –3.5

26 S(T)

Profit 19.5 17.5

Long put Long call Strangle

20 22.5 0 –0.5 –2 –2.5

17.5

25 S(T)

SUMMARY

- If the stock price at expiration is zero, the long put makes $21, but the long call loses $2, giving a gain of $19. As S(T) increases, the long call’s payoff remains parallel to the horizontal axis, but the long put drags the straddle’s payoff down at 45 degrees. The payoff cuts the x axis at the breakeven point $19, keeps declining, and reaches the nadir—a loss of $3.50. - For values of S(T) higher than $22.50, the long put’s payoff becomes parallel to the x axis, but the long call pulls the straddle’s payoff up at 45 degrees, which cuts the x axis again at the BEP = (22.50 + 2 + 1.50) = $26. For higher values of S(T), the payoff increases at 45 degrees, reflecting infinite profit potential. ■

If you reverse the preceding trades and sell a call and put, then you sell a straddle (establish a top straddle or a short straddle), a bet on low future volatility whose payoff is the mirror image of the bottom straddle across the horizontal axis.

Strangle

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■

A long strangle is created by buying options with different strike prices, where the strike price of the put is lower than the strike price of the call. Though it’s a volatility bet like a straddle, a strangle does have some differences: - Buy the OPSY 22.50 call for $2, and buy the OPSY 20 put for $0.50. Plot these two strategies in a profit diagram (see Figure 15.7). - At the origin, the profits for the strangle is 19.50 – 2 = $17.50. For higher values of S(T), it declines at a 45 degree angle until it reaches –$2.50 when S(T) is $20. For the stock price lying between $20 and $22.50, the profit is flat because both the call’s and put’s profits are parallel to the horizontal axis. Beyond $22.50, the long put’s profit stays flat, but the long call pulls up the strangle’s profit, which cuts the x axis at $25 and keeps increasing at a 45 degree angle, indicating an unlimited profit potential.

15.8

Summary

1. Options trading strategies’ performances are drawn in profit diagrams for European options with a single stock as the underlying. The basic building blocks for options trading are simple naked (or uncovered) strategies that are stand-alone positions buying or selling a single call or put option. 2. Hedged strategies combine options and stocks in a way that dampens the overall risk as compared to a naked option trade. They are usually covered strategies because they tend to back the option transaction with a stock that protects (or covers or collateralizes) it. Hedged strategies include covered call writing (long stock plus short call) and covered put buying (long stock plus long put). 3. Spread strategies combine several options of the same type (either both calls or both puts) on the same underlying but on different sides of the market. These two-sided hedges give a small profit if the underlying moves in one direction but suffer a small loss otherwise.

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4. Spreads come in three basic types: (1) vertical spreads involve options that have different strike prices but expire on the same date, (2) horizontal spreads involve options that have the same strike but different maturity dates, and (3) diagonal spreads involve options that differ both in terms of strike price and maturity date. 5. Several other categories of spread strategies can be created by adding options of the same type, with the same or different strike prices, to a spread strategy: - A ratio spread has the number of calls (or puts) bought different from the number of calls (or puts) sold. - A butterfly spread has four options of the same type (all calls or all puts) with three different strike prices: buy (sell) two options with extreme strike prices and write (purchase) two options with a middle strike price. - A condor spread has four options with four strike prices: purchase (write) the two extreme options and sell (buy) two options with strike prices in the middle. 6. Combination strategies combine options of different types on the same underlying stock and expiring on the same date, where both options are either purchased or written. A straddle combines a call and a put with the same strike price. A strangle combines a call and a put with different strike prices.

15.9

Cases

Cephalon Inc. (Harvard Business School Case 298116-PDF-ENG). The case intro-

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duces students to the use of equity derivatives as part of a risk-management strategy, examines the application of cash flow hedging in a corporate context, and explains the pricing of a derivative security with large jump risk. National Insurance Corp. (Harvard Business School Case 296036-PDF-ENG).

The case considers whether a major reinsurer should use exchange-traded insurance derivatives for managing catastrophic risk. Compensation at Level 3 Communications (Harvard

Business School Case 202084-PDF-ENG). The case analyzes a compensation plan that rewards managers for the firm’s performance only if the firm’s stock price movement exceeds a benchmark.

15.10 Questions and Problems The next eleven questions are based on the following data for a fictitiously named company, OOPS. ■

Current stock price S is $22.

■

Time to maturity T is six months.

■

Continuously compounded, risk-free interest rate r is 5 percent per year.

■

European options prices are given in the following table:

QUESTIONS AND PROBLEMS

Strike Price

Call Price

Put Price

K1 = $17.50

$5

$0.05

K2 = 20

3

0.75

K3 = 22.50

1.75

1.75

K4 = 25

0.75

3.50

15.1. Use options with strike K2 = $20.00: a. Draw a long call profit diagram. b. Draw a short call profit diagram. c. Draw a long put profit diagram. d. Draw a short put profit diagram. 15.2. Use options with strike K2 = $20.00: a. Draw a covered call (long stock plus short call) profit diagram. b. Give a reason why a trader might want to hold a covered call position. c. Explain the difference between a buy–write and an over-write strategy. 15.3. Use options with strike K2 = $20.00: a. Draw a covered put (long stock plus long put) profit diagram.

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b. Give a reason why a trader might want to hold a covered put position. c. Explain the difference between a married put and a protective put. 15.4. Use options with strike K3 = $22.50: a. Draw the profit diagram for a long call and a short put position for options

with the same strike and maturity date. b. Draw the profit diagram for a long stock and two short calls. 15.5 Using options with strike price K3 = $22.50, draw the profit diagram for a

long call, a short stock, and a short put position for options with the same strike and maturity date. What other investment has this profit diagram? 15.6. a. What is a bullish vertical spread? b. Draw a bullish vertical spread by trading put options with strike prices K2 =

$20 and K4 = $25. 15.7. a. What is a bearish vertical spread? b. Draw a bearish vertical spread by trading call options with strike prices

K1 = $17.50 and K3 = $22.50. 15.8 Draw a butterfly spread by going long calls with strike prices K1 = $17.50 and

K3 = $22.50 and selling short two calls with a strike price in the middle.

333

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15.9 Draw a condor spread by going long calls with strike prices K1 = $17.50 and

K4 = $25 and selling short two calls with each strike price in the middle. 15.10. a. What is the aim of a long (or bottom) straddle strategy? b. Create a long straddle by buying a call and a put with strike price K3 =

$22.50. 15.11. a. What is the aim of a short (or top) strangle strategy? b. Create a short strangle by writing a call with strike price K3 = $22.50 and

a put with strike price K1 = $20. The next five questions are based on the following options price data for Tel Tales Corporations (fictitious name), where the options expire on the same date in May. Draw profit diagrams in each case, clearly showing the stock price corresponding to zero profit, the maximum profit and loss, and so on. Stock

Strike (K)

Call Price

Put Price

$29

$25

$5

$1

30

2

3

35

1

6

15.12 A 3:1 reverse hedge (buy three May 30 calls and short the stock). Copyright © 2018. World Scientific Publishing Company Pte. Limited. All rights reserved.

15.13 A bullish spread (long call with strike price of $25 and short call with strike

price of $30). 15.14 A butterfly spread (long put with strike prices $25 and $35 and short two puts

in the middle). 15.15 A strangle (buy call with K = $30 and buy put with K = $25). 15.16 A straddle (buy a call and a put with K = $30). 15.17 Goldminers Inc. mines and refines ore and sells pure gold in the global market.

To raise funds, it sells a derivative security whose payoff is as follows: ■

Part of the security is a zero-coupon bond (which is sold at a discount and makes no interest payments) that pays a principal of $1,000 at maturity T.

■

Goldminers also pays an additional amount that is indexed to gold’s price (per ounce) at maturity S(T): 0 if S (T) ≤ $1,350 $30 [S (T) − 1,350] if $1,350 < S (T) ≤ $1,400 $1,500 if $1,400 < S (T) Analyze this derivative as a combination of bond and put options.

QUESTIONS AND PROBLEMS

15.18 a. What is a collar in the options market? b. How would you create a zero-cost collar? c. Why might a copper manufacturer find it useful to employ this strategy? 15.19 An insurance company has insured oil fields in the Middle East. Next, it

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purchases reinsurance to manage its “tail risk.” How can the reinsurance company hedge some of its risks by trading derivatives?

335

16 Option Relations 16.1 Introduction

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16.2 A Graphical Approach to Put–Call Parity 16.3 Put–Call Parity for European Options 16.4 Market Imperfections

16.5 Options Price Restrictions EXTENSION 16.3 The Superglue Argument

16.6 Early Exercise of American Options No Dividends Dividends and Early Exercise

EXTENSION 16.1 Put–Call Parity in Imperfect Markets

16.7 Summary

EXTENSION 16.2 Dividends and American Options

16.8 Cases 16.9 Questions and Problems

A GRAPHICAL APPROACH TO PUT–CA LL PARITY

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16.1

Introduction

Eager to find how the options on Palm Inc.’s stock were performing on their inaugural trading day, he switched on his computer, entered the Internet, and logged on to his online brokerage account. Like Alice in Lewis Carroll’s children’s classic Alice’s Adventures in Wonderland, he grew “curiouser and curiouser” and could hardly believe his eyes. Market-traded put options were trading $4 higher than the arbitrage-free price predicted by put–call parity. Filled with enthusiasm, he established a game plan based on the textbook prescription to exploit the mispricing, but he stumbled at the outset. He could not find Palm shares to borrow and sell short, a critical step in constructing the arbitrage. He (one of us) did not make millions that morning but acquired knowledge and, most important, a story to share with you! Later in this chapter, we will discuss the Palm arbitrage in greater detail. The last chapter analyzed various options trading strategies using their profit diagrams on the maturity date. Determining an option’s price prior to expiration is a far more difficult task. Although it’s easy to find the forward price using the no-arbitrage argument because of the forward’s linear payoff, the nonlinear option payoffs complicate the argument. In fact, to get an exact value, one needs to assume an evolution for the underlying stock price (which indicates how the stock price evolves through time), the most popular being a lognormal distribution. This is studied in subsequent chapters. But what can you learn about options properties without assuming such an evolution? This is the topic studied in this chapter. Here we study three different categories of options price relations. First, we establish put–call parity for European options. This fundamental relation stitches together the stock, a bond, and call and put options. Put–call parity is not new: financier Russell Sage used it to circumvent New York State usury laws in the late 1800s. Next, we examine some restrictions that options prices must satisfy under the no-arbitrage assumption. These provide the key steps to our final topic: the early exercise of American options. As early exercise features are embedded in various other financial securities, it is important to understand this topic well. Throughout this chapter, we maintain our standard assumptions: no market frictions, no credit risk, competitive and well-functioning markets, no intermediate cash flows (such as cash dividends), and no arbitrage. However, we will occasionally need to relax the assumption of “no intermediate cash flows” below.

16.2

A Graphical Approach to Put–Call Parity

In The Put-and-Call, published in London in 1902, Leonard R. Higgins describes the workings of London’s over-the-counter options market. In this book, he makes the following intriguing observation: that “a put can be turned into a call by buying all the stock” and that “a call can be turned into a put by selling all the stock.” As observed by Higgins, a clever arrangement of stocks and European options establishes put–call parity (PCP). Initially, we illustrate this result with payoff diagrams in the next example, and later by other methods.

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EXAMPLE 16.1: Establishing Put–Call–Parity by Payoff Diagrams (Replicating a Call) ■

We can replicate a European call option by trading the underlying stock, a European put option, and zero-coupon bonds. The options on the stock need to have a common strike price and a common expiration date. For simplicity we use the common options pricing data (COP data) from Chapter 15.

■

First, buy an OPSY 20 put option whose time T payoff diagram is drawn in Figure 16.1A. Next, buy one OPSY stock and place its payoff diagram below that of the put. Now create a third payoff diagram that sums the first two payoffs for each value of S(T). This payoff graph for the long put plus the stock starts at a value of K = $20 on the vertical axis. When S(T) exceeds $20, the graph rises at a 45 degree angle to the x axis.

■

For convenience, we redraw this portfolio payoff in Figure 16.1B. Draw another diagram underneath this depicting the payoff from shorting a bond with a future liability of $20. The present value of this cash flow today is $20B, where B is today’s price of a zero-coupon bond that pays a dollar after six months. The short bond’s payoff is a line parallel to the x axis at – $20.

■

Now sum the payoffs of the previous positions (Long put + Long stock) and short bonds. The portfolio (Long put + Long stock + Short bonds) replicates the payoff to a long call option with strike price K.

■

As the time T payoffs are the same, the law of one price equates the value of the two portfolios today:

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Long call = Long put + Long stock + Short bond

(16.1)

Expression (16.1) is put-call parity (PCP) for European options. We can view the left side, the call, as a market-traded asset and the portfolio on the right side as a synthetic call (a portfolio that synthetically constructs the traded call’s payoffs).

16.3

Put–Call Parity for European Options

We follow the data–model–arbitrage approach to establish put-call parity (PCP) for European options. As in the previous section, we consider a market-traded call and construct a synthetic call with identical future payoffs. By the law of one price, the traded and the synthetic call must have the same value today. This yields the PCP relation. In this example, instead of the law of one price, we use the equivalent nothing comes from nothing principle instead.

PUT–CALL PARITY FOR EUROPEAN OPTIONS

FIGURE 16.1A: Step One (PCP by Graphs): Long Put plus Long Stock Payoff 20

Long put

FIGURE 16.1B: Step Two (PCP by Graphs): (Long Put plus Long Stock) plus Short Bonds gives a Long Call Payoff Long put + Long stock 20

0 K = 20

S(T) 0

Payoff Long stock

20

S(T)

Payoff 0

S(T)

0 S(T)

Short bonds

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–20 Payoff Long put + Long stock

Payoff

20 Long call 0

0 20

S(T)

K = 20

S(T)

EXAMPLE 16.2: Put–Call Parity for European Options

The Data ■

Consider the COP data of section 15.4 of Chapter 15. - OPSY’s stock price S is $22.50 at time 0. - The European options on OPSY have a common strike price K = $20 and mature in T = 6 months. - Today’s put price p is $0.50. The call price c is to be determined.

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- The continuously compounded risk-free interest rate r is 5 percent per year. Today’s price of a zero-coupon bond paying $1 at time T is B = e−rT = $0.9753. - No dividends are paid on OPSY’s stock over the option’s life. ■

All trades are recorded in an arbitrage table. The first column of Table 16.1 gives the trade description and the second column records today’s cash flows. The expiration date payoffs are presented in the last two columns. The third column reports the payoffs when S(T) is less than or equal to $20, and the last column gives the payoffs when it’s greater than $20.

■

We first implement a stock purchase plan portfolio. - Buy the stock in the spot market for $22.50. The cash flow is recorded as −22.50 in the second column. The stock is worth S(T) after six months. Jot down S(T) in both the third and fourth columns. - Tap the bond market to borrow the present value of the strike price K = $20 by shorting zero coupon bonds. This cash flow reduces today’s payment. Record 0.9753 × 20 today and the liability -20 in the last two columns.

■

Next, using the options market, we create a synthetic short stock portfolio. - Sell one OPSY 20 call. Record this trade as +c in the second column. The call expires worthless if S(T) is less than or equal to $20. For higher values of S(T), the future payoff is a liability. Surrender the stock worth S(T) and receive $20 in return, for a cash flow of [20 − S(T)].

TABLE 16.1: Arbitrage Table for Put–Call Parity

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Portfolio

Today (Time 0) Cash Flow

Expiration Date (Time T) Cash Flow S (T) ≤ 20

20 < S (T)

Stock purchase plan Long stock

−22.50

S(T)

S(T)

Short bonds to borrow the present value of strike (K = $20, B = $0.9753)

0.9753 × 20

−20

−20

Short call

c

0

20 − S(T)

Long put

−0.50

20 − S(T)

0

Cash flows

−22.50 + 0.9753 × 20 + c −0.50

0

0

Synthetic short stock

- Buy an OPSY 20 put. The premium p = $0.50 is written as a negative cash flow today. If S(T) is less than or equal to $20, then the put holder receives $20 but has to surrender the stock worth S(T). Record the put payoff as [20 − S(T)] in this case. The put expires worthless when S(T) exceeds $20.

PUT–CALL PARITY FOR EUROPEAN OPTIONS

■

The first two trades (Long stock + Short bond) have the payoff [S(T) − 20] at expiration. The last two trades (Short call + Long put) create an equal but opposite payoff on the expiration date.

■

The final row of Table 16.1 considers net cash flows. The portfolio’s payoff is always zero on the expiration date.

■

The portfolio of stocks, bonds, and options created has a zero value for sure in the future. To prevent arbitrage, using the nothing comes from nothing principle, it must have zero value today. Consequently, −22.50 + 0.9753 × 20 + c − 0.50 = 0 or c = $3.49 This is the European call option’s arbitrage-free price.

The Model ■

Use symbols to generalize. Replace $22.50 with S, $0.9753 × 20 with B × K, and $0.50 with p to get the PCP for European options: − S + BK + c − p = 0

(16.2)

Arbitrage Profits

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■

A violation of PCP leads to arbitrage. Suppose you find an errant trader who quotes a call price of $4. As this is higher than the arbitrage-free price from PCP, create the portfolio in Table 16.1: buy the stock, short the bonds, buy the put, and sell the relatively overpriced call. This gives the cash flows −S + BK + c − p = −22.50 + 0.9753 × 20 + 4 − 0.50 = $0.51 as your immediate arbitrage profit.

■

If the trader quotes $3 instead, reverse the trades and make $0.49 as an arbitrage profit.

We use expression (16.2) to formally state our result.

RESULT 16.1 Put–Call Parity for European Options c = p + S − BK

(16.3)

where c and p are today’s prices of European calls and puts, respectively, that have strike price K and expiration date T, B is today’s price of a zero-coupon bond that pays a dollar at time T, and S is today’s price of the underlying stock.

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PCP is a useful tool for understanding how to use options in forming trading strategies. We start with the put because it is the easiest option to understand. Indeed, a put option is like an insurance policy because it insures the value of the underlying stock at its strike price. With this insight, we can rewrite PCP as Call = Insurance + Stock + Borrowing

(16.4)

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We see that a call option is equivalent to buying the stock on margin (by borrowing) and protecting the stock purchase with an insurance policy. This intuition can help us to understand how to trade on information using options. For example, suppose that you are the only person in the market who knows that the stock is going to rise above the strike price. (Note: this is a thought experiment—we are not considering issues like whether trading on inside information will attract jail time!) What is the best way to trade on this information: buying a call? No. This is the wrong answer because by buying the call, you are also paying for the put, which is insurance on the stock that you do not need. Recall that you are certain that the stock price will rise. In this case, the optimal strategy is to buy the call and sell the put. Using PCP again, we also see that this is equivalent to buying the stock and financing the purchase with borrowed cash, which creates leverage (called gearing in Europe). Now suppose that you expect that the stock is likely to increase but you are unsure. In this case, is buying the call the best strategy? Maybe. The answer really depends on your risk tolerance. Pay for the put if you like the downside insurance protection but not otherwise.

16.4

Market Imperfections

Market imperfections introduce difficulties, but they may also create opportunities. Merton Miller’s observation that financial securities can help overcome rules and regulations becomes pertinent. Options and PCP can provide tools to work around these imperfections. In the article “The Ancient Roots of Modern Financial Innovation: The Early History of Regulatory Arbitrage,” Michael Knoll (2008) cited ancient examples of merchants using PCP to get around restrictions on interest payments. For example, two millennia back, Israelite financiers used PCP in this way. English financiers did the same five hundred years ago, and interestingly, the transaction they devised led to the development of the modern mortgage. These examples illustrate tax or regulatory arbitrage, which involves unpackaging and rebundling cash flows with the aim of profiting from otherwise prohibited transactions. Extension 16.1 discusses an example of PCP in action. One can tinker with PCP to generate two extensions: (1) to accommodate dividends and (2) for American options. These extensions are shown in Extension 16.1 and 16.2.

MARKET IMPERFECTIONS

343

EXTENSION 16.1: Put–Call Parity in Imperfect Markets

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PCP Violations for Palm Shares Sometimes PCP fails because of market imperfections. This happened with Palm Inc. options in an incident discussed at the chapter’s start. On March 2, 2000, 3Com did an equity carve-out of its subsidiary Palm, which had made the world’s first successful handheld computer, the Palm Pilot, in 1992. Accordingly, 3Com sold 5 percent of Palm shares to the general public in an initial public offering and declared that later in the year, it would distribute the remaining shares in a spin-off to its existing shareholders (who would get 1.525 shares of Palm for every share of 3Com they held). The mispricing that resulted in the equity and options markets got extensive press coverage and academic study (see Lamont and Thaler 2003). A hot company during the days when tech stocks led the dotcom stock market craze, Palm had an unbelievable initial public offering (IPO). Its stock closed at the end of the first day’s trading at $95.06 per share. This implied that 3Com’s stock price, which would equal 1.5 Palm shares plus the value of 3Com’s other assets and businesses was $63 per share, or – $22 billion. The mishmash confused options traders. They adopted strategies like writing call options and short selling 3Com stock to buy Palm stock, whose market capitalization then exceeded the parent company’s valuation! “There is absolutely no rationale behind it,” interjected a trader. Predictably, Palm started declining after the IPO.1 When options began trading on March 16, 2000, Palm stock traded around $55. For options maturing in one month, the at-the-money (strike price K = $55) call price c was $5 and the put price p was $9. These numbers are illustrative. The bid/ask spread was large, and the market was in a state of confusion, making it hard to find accurate prices. A zero-coupon bond price paying $1 after one month was worth $0.995. A quick check of put–call parity suggests Synthetic put price = Call price + Present value of strike –Stock price = 5 + 0.995 × 55 − 55 = $4.725 And yet the price of the market-traded put was $9. Isn’t this an arbitrageur’s dream come true? Wouldn’t this mean that if you sell the market-traded put and buy the synthetic put (see expression [16.2]), you can immediately capture an arbitrage profit of 9 − 4.725 = $4.275 in one trade? Repeat a million times, and you become a millionaire four times over! Unfortunately, a market imperfection disrupted this trade. To create the arbitrage portfolio, an “arbitrageur” needs to sell the relatively overpriced market-traded put and “buy” the relatively underpriced synthetic put via buying the call, shorting the stock, and lending the present value of the strike. In this circumstance, however, the broker could not supply the Palm shares for short selling (by borrowing them from another investor). The scarcity of shares for short selling prevented the realization of the arbitrage profits and led to the divergence between theory and practice. The situation did not last long. The price anomaly disappeared within days; however, a lesson was learned. Paper profits can differ from actual realizable profits, and market imperfections do matter. 1

Palm IPO Soars, Then Retreats a Bit, Pushing Traders to Unwind Options in Parent 3Com,” W allStreetJournal, March 3, 2000.

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EXTENSION 16.2: Dividends and American Options

Put–Call Parity Adjusted for Dividends The parallels are eerie. One can easily modify the basic PCP by using the tools developed in Chapter 12 for adjusting the cost-of-carry model for dividends. The intuition remains the same as before. A dividend lowers the cost of a stock purchase, and replacing the stock price with the “stock price net of all dividends over the life of the derivative” modifies the formula. Assuming that a fixed-dollar dividend div is paid on a known future date t1 and B1 is today’s price of a zero coupon bond maturing on the ex-dividend date (recall Result 12.1a), put–call parity for European options (with a known dollar dividend) can be written as c = p + (S − B1 div) − BK

(1)

As before, this can be easily generalized to accommodate multiple dividends. For options on indexes, we can use the insights from Result 12.1b. If the underlying pays dividends at a continuous rate 𝛿, then put–call parity for European options (with a known dividend yield) can be written as c = p + e−𝛿T S − e−rT K

(2)

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A Put–Call Parity Inequality for American Options The basic PCP does not work for American options because the option may get exercised before the expiration date. However, a PCP inequality for American options does hold. For these inequalities, we add the assumption that interest rates are nonnegative, or equivalently, that a zero-coupon’s price is always less than or equal to 1.

EXT. 16.2 EX. 1: A Relation between American Calls and Puts

The Data ■

The stock price is S today. Assume that the stock pays no dividends over the life of the options.

■

American calls and puts worth cA and pA , respectively, trade on the stock. They have the same strike price K and expire at time T but may get exercised early at time t.

■

Interest is compounded continuously at the risk-free rate r percent per year. You can invest in zero-coupon bonds or a money market account (mma). B is today’s price and B(t) is the price at time t of zeros that mature at time T. An investment of $1 in a mma today grows to [1 + R(t)] on the intermediate (exercise) date t and to (1 + R) on the expiration date.

When the Portfolio Involves a Long Put (and Short Call) ■

Create a portfolio similar to the basic PCP: buy one put, write one call, buy one stock, and short bonds to borrow the present value of the strike. Today’s portfolio value is pA − cA + S − BK

(3)

MARKET IMPERFECTIONS

345

You decide when to exercise the put. Your counterparty decides when to exercise the American call. Thus two things can happen in the future: the holder of the American call exercises early, or she does not. Your response depends on Long’s decision on the call. ■

If the American call is not exercised early, then the American call becomes equivalent to the European call worth c. You can write cA = c. Now, if you don’t exercise your American put early, then it behaves like a European put, which means pA = p. With cA = c and pA = p, expression (3) becomes equal to 0 by PCP for European options. But you can exercise your American put before it expires. This flexibility makes the American put worth at least as much as the European put. Thus (3) becomes pA − cA + S − BK ≥ p − c + S − BK = 0

■

(4a)

If your counterparty exercises the American call early, then portfolio (3) has a payoff on the exercise date t (when you buy back the bonds by paying B[t] for each zero): pA (t) − [S (t) − K] + S (t) − B (t) K = pA (t) + [1 − B (t)] K ≥ 0

(4b)

This is nonnegative because a put cannot have a negative value and B(t) is worth less than or equal to a dollar. ■

Thus the portfolio (pA − cA + S − BK) has a value greater than or equal to zero. Combining (4a) and (4b) and moving cA to the right side, we get our first inequality

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pA + S − BK ≥ cA

(5)

When the Portfolio Involves a Short Put (and Long Call) ■

Create the portfolio: sell the put, buy the call, short the stock, and invest K dollars in a mma. Notice that this differs from the basic PCP portfolio because more dollars are kept in case the put is exercised early. This portfolio’s value today is − pA + cA − S + K (6) Here the buyer determines whether to exercise the American put. As before, we will act in response to Long’s decision.

■

If there is no early exercise, then the American put is the same as the European put, pA = p. If you don’t exercise the American call early, then it behaves like the European call, which means cA = c. With cA = c and pA = p, (6) becomes equal to KR on the maturity date (you can verify this in a payoff table). Because the last quantity is greater than zero, (6) cannot be less than zero today: − pA + cA − S + K ≥ 0

(7a)

Because you can exercise your American call early, this makes it even more valuable, which reinforces the relationship in (7a).

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If the buyer exercises early the American put, then portfolio (6) has a payoff on the exercise date t (when K has grown to K[1 + R(t)]): − [K − S (t)] + cA − S (t) + K [1 + R (t)] = KR (t) + cA ≥ 0

(7b)

because both the call and R(t) are nonnegative. ■

Thus the portfolio (− pA + cA − S + K) cannot have a negative value today. Combining (7a) and (7b) and rearranging terms, we get our second inequality: cA ≥ pA + S − K

■

(8)

For the PCP inequality for American options, combining (5) and (8), we get the relation between the stock, the bond, and the American options pA + S − K ≤ cA ≤ pA + S − BK

16.5

(9)

Options Price Restrictions

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This section explores various options price restrictions for several reasons. ■

They hone intuition and enhance understanding.

■

They generate arbitrage opportunities when the bounds are violated.

■

Just as you stretch before you sprint, a discussion of options price relations forms the stepping-stone to understanding early exercise of options, which is the final topic of this chapter.

To facilitate understanding, we establish the options price restrictions in several different ways: (1) a diagrammatic approach with profit diagrams; (2) an arbitrage table approach, where cash flows are recorded in a table and no arbitrage yields the result; and (3) a superglue argument, which is our name for comparing the original security with a “restricted” but otherwise identical security and then proving the result by arguing that “less cannot be worth more.” Why “superglue?” Read extension 16.3. We begin with the simple and then move to the complex. We use our standard notation and assumptions, except one: we allow the underlying stock to pay dividends. This is an important extension because most stocks pay dividends.

OPTIONS PRICE RESTRICTIONS

RESULT 16.2 American Options Are Worth More Than European Options 2a. 2b.

cA ≥ c ≥ 0 pA ≥ p ≥ 0

(16.5a) (16.5b)

This result holds because European and American options are identical, except for their exercise features. European options can only be exercised at expiration, whereas American options can be exercised anytime. As “more cannot be worth less,” an American option can never be priced less than an otherwise identical European option. This simple yet robust principle underlies the superglue argument, which is used later to prove some results (see Extension 16.2). Of course, an option’s price must be nonnegative because the holder can always discard the option without exercising it. The next two results restate the option price bounds established earlier in Chapter 5.

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RESULT 16.3 Call Price Boundaries 3a. An American call’s price is less than or equal to the stock price, that is, cA ≤ S

(16.6)

3b. When the stock price is zero, the call price is also zero.

Result 16.3 also holds for European calls because they are less valuable than American calls. Notice that an American call’s price cannot fall below the boundary condition stated earlier in Chapter 5. These are depicted in the first diagram in Figure 16.2, which reproduces Figure 5.3. Result 16.3b follows by setting S = 0 in Result 16.3a and recognizing that the call’s value must always be nonnegative.

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RESULT 16.4 Put Price Boundaries An American put’s price is less than or equal to the strike price K, and a European put’s price is less than or equal to the present value of the strike price: 4a. pA ≤ K (16.7a) 4b. p ≤ BK (16.7b) where B is today’s price of a zero-coupon bond that pays $1 at the option’s maturity.

A put option entitles the holder to receive the strike price, but he has to surrender the stock in return. Thus the put price cannot exceed the strike price, which forms an upper bound. Because a European put’s maximum payoff is the strike and only on the expiration date, today’s put price must be less than or equal to the present value of the strike price.

RESULT 16.5

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Call Price Bound before Expiration Before expiration, a European call’s price is greater than or equal to the larger of the stock price minus the present value of the strike or zero: c ≥ max (S − BK, 0)

(16.8)

We will prove this result with payoff diagrams, by an arbitrage table, and using the superglue argument (see Extension 16.3). The result also holds for American options, which are more valuable than otherwise identical European options. Figure 16.3 shows two payoff diagrams on the expiration date. The first diagram is for a stock purchase plan that consists of buying the stock and shorting K zerocoupon bonds. The second diagram, which is placed beneath the first, is for a long call. Visual inspection shows that the stock purchase plan pays off less than the long call. A consideration of today’s value of these two payoffs establishes that c ≥S − BK. Because the call’s value must also be nonnegative, we get expression (16.8). Our next example uses a variant of our data–model–arbitrage approach to establish the relation.

OPTIONS PRICE RESTRICTIONS

FIGURE 16.2: Price Bounds for American Calls and Puts Prices

Upper Bound Lower Bound Call prices in here

0

45° K

S, stock price

Prices Upper Bound

K Put prices in here

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0

K

Lower Bound S, stock price

EXAMPLE 16.3: Establishing Result 16.5 with an Arbitrage Table

The Data, Arbitrage, and Diagram ■

Consider the data from Example 16.2 with a slight modification. OPSY’s stock price S is $22.50, the six-month $20 strike European call price c is $2 and the put price p is $0.50, and the zero-coupon bond price B is $0.975310.

■

Does Result 16.5 hold? We have S − BK = 22.50 − 0.9753 × 20 = $2.9938. This is greater than the call price of $2, and the inequality in (16.8) is violated. Transferring the right side terms to the left side, we get the expression c − S + BK = 2 − 2.9938 = −$0.9938.

■

Buy low, sell high. Create an arbitrage portfolio by buying the underpriced call and selling the overpriced portfolio (S − BK) by shorting the stock and buying the bonds. Record these cash flows in Table 16.2. You immediately get $0.9938. At expiration, the portfolio has a 0 payoff when the call is in-the-money and a nonnegative payoff of [20 − S(T)] otherwise.

■

You can also see this arbitrage possibility by drawing the profit diagram for the portfolio in Table 16.2. Take $0.9938 and invest it in bonds to get $0.0190 after six months. The portfolio payoff after six months is [20 − S(T)] when the call is out-of-the-money, and 0 otherwise. Adding $1.02 (after rounding) to this shifts the whole curve up (see the diagram in Figure 16.4). This confirms the arbitrage opportunity.

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To establish the result, one simply replaces the numbers with algebraic symbols. Writing S = $22.50, K = $20, B = $0.975310, and c = $2, the last row in the second column of Table 16.2 shows that − c + S − BK = 0.9938. You can generalize the last row, as c < S − BK leads to a cash inflow today and a nonnegative payoff [K − S(T)] when S(T) ≤ K, and 0 otherwise. This is an unstable situation. To prevent arbitrage, we must have c ≥ S − BK (16.9) This is Result 16.5 when combined with the fact that c ≥0. Figure 16.4 also demonstrates an easy way to identify an arbitrage portfolio by drawing a profit diagram and checking to see whether you can make arbitrage profits.

FIGURE 16.3: The Call Price Must Exceed the Stock Price Minus the Present Value of the Strike Price (Result 16.5) Payoff

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Long stock

0 K = 20

S(T)

–20 Payoff Long call

0 K

S(T)

OPTIONS PRICE RESTRICTIONS

TABLE 16.2: Arbitrage Table Showing How a Violation of c ≥ S – BK (Result 16.5) Creates an Arbitrage Opportunity Portfolio

Today (Time 0) Cash Flow

S(T) ≤ 20

20 < S(T)

Buy call

−2

0

S(T) − 20

Short sell stock

22.5

−S(T)

−S(T)

−0.975310 × 20

20

20

0.9938

20 − S(T)

0

Buy bonds to lend the present value of strike (K = $20, B = $0.975310) Cash flows

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Expiration Date (Time T) Cash Flow

■

If the profit graph for a portfolio lies entirely above (or with some portions of it lying along) the horizontal axis, then you have created an arbitrage portfolio.

■

If the profit graph lies entirely below (or with some portions of it lying along) the horizontal axis, just reverse the trades to capture arbitrage profits.

■

A profit graph that crosses the axis leads to both trading profits and losses, which can never represent an arbitrage opportunity.

FIGURE 16.4: Profit Diagram Showing an Arbitrage Opportunity When the Call Price Is Less Than the Stock Price Minus the Present Value of the Strike Price (Violation of Result 16.5) Profit

21.02

1.02 0 K = 20

S(T)

351

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CHAPTER 16: OPTION RELATIONS

Result 16.5 has its counterpart for put options.

RESULT 16.6 Put Price Bound before Expiration 6a. Before expiration, a European put’s value must be greater than or equal to the larger of the present value of the strike price minus the stock price or zero: p ≥ max (BK − S, 0) (16.10a) 6b. Before expiration, an American put’s value must be greater than or equal to its intrinsic value: pA ≥ max (K − S, 0) (16.10b)

Result 16.6a follows easily using the superglue argument (see Extension 16.3). Result 16.6b follows from the boundary condition because you can always exercise an American put early and collect the strike price by surrendering the stock.

RESULT 16.7

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Relation between Options with Different Strike Prices 7a. The lower the strike price, the more valuable the European call: c (K2 ) ≤ c (K1 ) for K1 < K2

(16.11)

7b. The higher the strike price, the more valuable the European put: p (K1 ) ≤ p (K2 ) for K1 < K2

(16.12)

Both these results hold for American options as well.

Extension 16.3 proves several of these results using the superglue argument.

EXTENSION 16.3: The Superglue Argument Remember superglue, the unusually strong adhesive that advertisements claim can hang a car and lift two thousand pounds per square inch? Our use of superglue is different—we take the idea, but not the product! Consider two identical securities, which we name Sec. We attach a “restriction” to one of these securities, something

OPTIONS PRICE RESTRICTIONS

353

unfavorable or adverse that removes one of its provisions, and assume that it gets fastened with superglue (see Ext. 16.3 Fig. 1). As such, the restriction is permanent, and the security’s provisions are changed forever.

EXT. 16.3 FIG. 1: The Superglue Argument for Options SecR GLU

Sec E

Restriction

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Security

Security + Restriction

SecR is identical to Sec except for a restriction that is attached by superglue. Then, price of SecR < _ price of Sec.

The essence of the superglue argument is this: consider a security Sec and an otherwise identical security with the restriction attached, which we call SecR (short for SecurityRESTRICTED ). A long position in SecR cannot have a greater value than Sec because less cannot be worth more, that is, Price of SecR ≤ Price of Sec. This is a no-arbitrage condition. If it happens otherwise, then buy the security, add the restriction (with superglue at no cost), and sell it for more to make arbitrage profits. We can use this principle to derive all of our options price bounds and relations. We illustrate just a few of these below. Consider the proof of result 16.2 (directly below). Let Sec be an American option. Next, we attach a restriction to this option with superglue: “it cannot be exercised early.” Then, by no arbitrage, we know that the price of the restricted security—equivalent to a European call—must be less than the price of the unrestricted security—the American call. (Figure 16.5 illustrates this approach).

Result 16.2 American Options Are Worth More Than European Options Proof: Sec: American option. SecR : American option + Restriction “No early exercise allowed” Because SecR has become an otherwise identical European option, the superglue argument gives: European option price ≤ American option price.

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CHAPTER 16: OPTION RELATIONS

Result 16.3b cA ≤ S Proof: Sec: Stock SecR : Stock + Restriction “Pay an additional K dollars to maintain possession, otherwise throw away, and the decision must be made before time T.” Of course, with this restriction attached, the restricted stock is an American call. Hence American call price ≤ Stock price.

Result 16.5 c ≥ S – BK Proof: Sec: European call. SecR : European call + Restriction “Always exercise the call even if it is out-of-the money.” Today’s price of SecR is S − BK. The superglue argument gives the result. Result 16.2 tells us that this argument also holds for American calls.

Result 16.7a c(K1 ) ≥ c(K2 ) for K1 < K2

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Sec: European call with strike price K1 . SecR : European call with strike price K1 + Restriction “Must pay the extra amount (K2 − K1 ) in case of exercise.” The restricted security is equivalent to a European call with strike price K2 . The superglue argument gives the result. A similar argument holds for American calls.

EXAMPLE 16.4: Result 16.7b by the Modified Data–Model–Arbitrage Approach ■

Consider the COP data from chapter 15 with minor modifications. Puts trade on OPSY that mature in six months. The premiums are p(20) = $2 for strike price K1 = $20 and p(22.50) = $1.50 for K2 = $22.50. These numbers violate inequality (16.12). We use the arbitrage opportunity in the data to justify our model, which is relation (16.12).

When Both Puts Are European ■

As shown in Table 16.3, we create the arbitrage portfolio by selling the overpriced OPSY 20 put and buying the underpriced OPSY 22.50 put. This gives a cash flow of + 2 − 1.50 = $0.50 today. The puts’ payoffs on the expiration date depend on the stock price. - If S(T) is less than or equal to $20, the short put has a payoff of −[20 − S(T)] and the long put has a payoff of [22.50 − S(T)]. This gives a payoff of $2.50. - For 20 < S(T) ≤ 22.50, the OPSY 20 put has a 0 payoff while the long OPSY 22.50 put has a payoff of [22.50 − S(T)]. For this range for S(T), the payoff is nonnegative. - When S(T) is greater than $22.50, both options are out-of-the-money and have 0 payoffs.

■

We showed that p (20) = $2 > p (22.50) = $1.50

355

OPTIONS PRICE RESTRICTIONS

leads to an arbitrage opportunity. The no-arbitrage requirement dictates that p(20) ≤ p(22.50). ■

Replace numbers with symbols and you get the result p (K1 ) ≤ p (K2 ) for K1 < K2

When Both Puts Are American ■

Suppose the mispricing holds and the prices reflect those for American puts instead. As with European puts, create an arbitrage portfolio by selling the OPSY 20 American put and buying the OPSY 22.50 American put. This gives $0.50 today. In the future, if the short put is not exercised against you, hold on to the long put. The payoff on the expiration date is given in Table 16.3. By contrast, if the OPSY 20 put is exercised against you, exercise the OPSY 22.50 put to eliminate the stock position, and you are left with $2.50 on the exercise date. This trade is an arbitrage opportunity. Hence, the only American put prices consistent with no arbitrage must satisfy result 16.7b.

TABLE 16.3: Arbitrage Table for Result 16.7b: The Higher the Strike Price, the More Valuable Is the Put Expiration Date (Time T) Cash Flow

Portfolio

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Sell 20 put Buy 22.50 put Cash flows

S (T) ≤ 20

20 2.50 because B < 1. Given these thought experiments, we now better understand Result 16.10a. It says that you can make more profits by selling the OPSY call rather than by exercising it. Thus, if no dividends are paid to the stockholder over the option’s life, then the call option is always worth more alive than dead. Do not exercise such a call: sell it. Sell it! SELL IT!! Puts, though, have a different story. Unlike calls that have an unbounded profit potential, put payoffs are capped by the strike price, the maximum payoff possible. This leads to Result 16.10b. To understand why Result 16.10b is true, consider the following hypothetical example. Consider a six-month original maturity put on OPSY that began two months back. If OPSY’s stock price is S(t) = $0.0001 today, an immediate exercise gives K − S(t) = (20 − 0.0001) = $19.9999. If the interest rate r is 6 percent per year, investing the proceeds in a money market account over the next day yields [K − S (t)] erT = [19.9999] e0.06 × (1/365) = $20.0032

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Waiting one day and then deciding to exercise yields at most $20, and if one waits, the stock price may rise and even less of a payoff may occur. So early exercise is optimal today. Of course, this is a pathological example selected to make our point, but it does prove that early exercise is sometimes optimal if the stock price is low enough and interest rates are high enough. This pathological example also illustrates another observation. It is easy to believe that as the stock price rises, there is some stock price S* where you are just indifferent between exercising today or not. The S* just balances the interest earned on the proceeds of early exercise with the chance of the stock price declining more (see Figure 16.6). Such an S* does in fact exist. Unfortunately, to prove this observation and to determine the S* precisely, we need a model for the stock price evolution. This analysis will be pursued in chapter 18. For now, we can only prove Result 16.10b.

Dividends and Early Exercise Dividends change the optimal exercise decision for American options. Recall that when dividends are paid, the stock price falls. We characterized this dividend-induced price drop as Result 3.1:3 SCDIV = SXDIV + div (3.1) where SCDIV is stock price cum-dividend, SXDIV is stock price ex-dividend, and div is the amount of dividend. A natural place to investigate early exercise is around the time a stock goes ex-dividend. This leads to our next result.

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FIGURE 16.6: Illustration of Result 16.10b Prices

K = 20 Stock price S* 0 t T Early exercise

Time

At S*, one is indifferent between exercising or not. Lower stock prices give higher proceeds from early exercise.

3

Exceptions to this statement can happen. We need some technical assumptions about the process underlying stock price changes. The statement in (3.1) holds, for example, if the stock price process follows a lognormal distribution, which underlies the Black–Scholes–Merton model. It does not hold, for example, if the stock price process allows discrete jumps and the jump happens at the same time as the stock goes ex-dividend; see Heath and Jarrow (1988) for a formal discussion.

EARLY EXERCISE OF AMERICAN OPTIONS

RESULT 16.11 Optimal Times for Exercising American Calls The only times when it may be optimal to exercise early is just before a stock goes ex-dividend.

The next example establishes this result with an application of Result 16.5.

EXAMPLE 16.6: Establishing Result 16.11 ■

Consider a six-month OPSY 20 call. Let the next dividend div1 be paid at time t1 , which is two months from today (time 0). Should you exercise this call at a time t, anytime over the next two months?

■

Consider the two possible strategies: early exercise at time t or wait and exercise at time t1 . If you exercise at time t, the value of your position is S(t) − 20, but if you exercise an instant before OPSY goes ex-dividend at time t1 , even if the option is out-of-the-money, then you get SCDIV − 20. The present value at time t of waiting until time t1 to exercise is

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S (t) − 20B where B is the time t price of a zero-coupon bond that pays $1 at time t1 . ■

Because B is less than 1, the second strategy gives you more value than the first. Thus exercising the American call just before the stock goes ex-dividend is better than exercising it anytime between now and the first dividend date.

■

This result follows because exercising earlier than these ex-dividend dates leads to an early surrender of the strike price—an action that loses interest but does not generate any additional benefit. Hence, if the call is exercised early to obtain the dividend on the stock, it will be exercised just an instant before the stock goes ex-dividend. This is depicted in Figure 16.7. See Jarrow and Turnbull (2000) and other finance textbooks for a discussion of how big the dividend needs to be to trigger early exercise.

And what about developing early exercise results for puts? Puts can get exercised early even without dividends. As dividends would cause the stock price to drop further, their presence is likely to delay early exercise. To get more precise insight, we need to model the stock price evolution. Again, we address this in later chapters.

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FIGURE 16.7: Timeline for Exercise of American Options If exercise call, exercise at these points

Option starts

Ex-div (div1)

Ex-div (div2)

Expiration

Exercise put anywhere If exercising an American call early, exercise just before the stock goes ex-dividend. An American put may be exercised anytime (but dividends tend to delay exercise).

16.7

Summary

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1. Put–call parity for European options (see Result 16.1) states that the call price plus the present value of strike equals the stock price plus the put price, or c + BK = S + p, where B is today’s zero-coupon bond price paying a dollar on the options’ common maturity date and K is the common strike price. 2. PCP has a number of applications and uses: - PCP can be used to create synthetically an option, a stock, or a bond. - PCP can be easily modified to adjust for known dividends. 3. Option prices satisfy restrictions that can be derived using the no-arbitrage principle. These theorems hone intuition and enhance understanding, lead to arbitrage profits in case of violations, and form a stepping-stone to understanding the early exercise of options. 4. Options price restrictions can be established in several different ways: (1) a diagrammatic approach with profit diagrams, (2) an arbitrage table approach, where cash flows are recorded in a table and no arbitrage yields the result, and (3) a superglue approach, which is our name for comparing the original security with a “restricted” but otherwise identical security and then proving the result by arguing that “less cannot be worth more.” 5. Some options price properties for stocks that may pay dividends are as follows: - Result 16.2—American options are worth more than European options: cA ≥ c and pA ≥ p. - Result 16.3—Call price boundaries 3a An American call’s price is less than or equal to the stock price: cA ≤ S. 3b When the stock price is zero, the call price is also zero.

SUMMARY

- Result 16.4—Put price boundaries 4a An American put’s price is less than or equal to the strike price K: pA ≤ K. 4b A European put’s price is less than or equal to the present value of the strike price: p ≤ BK. - Result 16.5—Before expiration, a European call price must exceed the stock price minus the present value of the strike price: c ≥ S − BK. - Result 16.6a—Before expiration, a European put must be greater than or equal to the present value of the strike price minus the stock price: p ≥ BK − S. - Result 16.6b—Before expiration, the value of an American put must be greater than or equal to its intrinsic value: pA ≥ K − S. - Result 16.7a—The lower the strike price, the more valuable is the European call: c(K1 ) ≥ c(K2 ) for K1 < K2 . - Result 16.7b—The higher the strike price, the more valuable is the European put: p(K1 ) ≤ p(K2 ) for K1 < K2 . These results hold for American options as well. - Result 16.8—Option prices and time to maturity: the longer the time until expiration, the greater the price of an American call or a put.

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8a cA (T1 ) ≤ cA (T2 ) for T1 < T2 . 8b pA (T1 ) ≤ pA (T2 ) for T1 < T2 . However, these results need not hold for European calls or European puts. 6. Early-exercise considerations begin with the following: Result 16.9a—Exercise an American call the first time its price is equal to the stock price minus the strike price: cA = S − K. Result 16.9b—Exercise an American put the first time its price is equal to the strike price minus the stock price: pA = K − S. 7. Dividends play a critical role in early-exercise decisions: - Assume the stock pays no dividends over the option’s life: Result 16.10a—The American call becomes identical to a European call. It should never be exercised early. Result 16.10b—The American put should be exercised if the stock price is small enough relative to the strike price and the time to maturity. - Absent market frictions, dividends cause the stock price to drop by the amount of the dividend (Result 3.1). This is used to establish the following: Result 16.11—The only times when it may be optimal to exercise an American call early is just before a stock goes ex-dividend. Puts can get exercised early even without dividends, but dividends are likely to delay early exercise.

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16.8

Cases

Boston Properties (A and B) (Harvard Business School Cases 211018 and 211041-

PDF-ENG). The case introduces options pricing, payoff diagrams, and the law of one price and explains arbitrage as well as no-arbitrage bounds. Smith, Barney, Harris Upham and Co. Inc. (Darden School of Business Case

UV0074-PDF-ENG, Harvard Business Publishing). The case approaches put–call parity from the trader’s perspective and examines the practical aspects of doing arbitrage in the options markets. Sleepless in L. A. (Richard Ivey School of Business Foundation Case 905N11-

PDF-ENG, Harvard Business Publishing). The case discusses the Black–Scholes– Merton model for options pricing, the concept of implied volatility, and put–call parity. It also shows how options pricing can be used to value corporate liabilities of a financially distressed company.

16.9

Questions and Problems

The next two questions are based on the following data for European options: Call price = $5, risk-free continuously compounded interest rate r = 5 percent per year, stock price S = $55, strike price K = $55, time to maturity T = 1 month. 16.1. If the put price p = $9, show how to capture arbitrage profits in this market. 16.2. Suppose short selling of stocks is not allowed in this market. Can you still

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make arbitrage profits? Explain your answer. 16.3. Do the following data satisfy put–call parity for European options? If they

don’t, show how you can create a portfolio to generate arbitrage profits. Call price = $6, put price = $3, stock price S = $102, strike price K = $100, time to maturity T = 3 months, and risk-free continuously compounded interest rate r = 5 percent per year. 16.4. How can you adjust put–call parity for known dividends on a single known

date? 16.5. Prove put–call parity for European options in the case of a single known

dividend: c + PV (Div) + Ke−rT = p + S where S is stock price, K is strike price, T is maturity date for the option, r is risk-free interest rate, c is European call price, p is European put price, and PV(Div) is the present value of dividends. 16.6. Using put–call parity, given c = $2, PV(Div) = $1, p = $1, S = K = $100,

r = 0.05 per year, and T = 0.25 years, can you make arbitrage profits? Explain. 16.7. Explain the relation between a put option (with strike price K and maturity

date T years from today) and a T-period insurance policy that insures the stock for K dollars.

QUESTIONS AND PROBLEMS

16.8. Different countries, different customs, different market practices! Suppose you

go to a country where traders with inside information can easily trade. There, you have inside information that the stock price is going to increase for sure (no chance that it will decline), and it is legal to trade on this information. What is the best strategy to use? 16.9. Does put–call parity always hold in financial markets? If not, give a few reasons

why it may not hold. 16.10. On the day options on Palm began trading, the share prices grossly violated

the put–call parity. Describe and explain why this happened. 16.11. Can you link Russell Sage’s actions with a distinguished economist’s view on

what drives financial innovation? 16.12. Is it true that the lower the exercise price, the more valuable the call? Explain

your answer. 16.13. Is a European put on the same stock with the same maturity worth more or

less if the strike price increases? Explain your answer. 16.14. Is it true that the more the time until expiration, the less valuable an American

put? Explain your answer. 16.15. Can you make arbitrage profits from the following European call prices?

If so, give two such examples of arbitrage, neatly showing the portfolio construction as well as the various cash flows. The stock price is $40.

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Strike Price

Expiration Month April

July

September

35

1

6

3

40

2

5

6

16.16. The following prices are given for American put options on a stock whose

current price is $100: Strike Price

Expiration Month March

June

95

1

11

100

7

5

Construct three portfolios for making arbitrage profits, showing the cash flows from each portfolio.

365

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16.17. The following prices are given for American call options on a stock whose

current price is $100: Strike Price

Expiration Month March

June

95

11

10

100

8

8

105

2

5

Construct three portfolios for making arbitrage profits, showing the cash flows coming from each portfolio. 16.18. If an American call is written on a stock that never pays a dividend, would

you ever exercise the call option early? Explain your answer. 16.19. If an American put is written on a stock that never pays a dividend, would

you ever exercise the put option early? Explain your answer. 16.20. a. Is there a simple rule of thumb that you can use to know when to exercise

an American call early? If yes, explain the rule. b. Is there a simple rule of thumb that you can use to know when to exercise

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an American put early? If yes, explain the rule.

17

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Single-Period Binomial Model 17.1 Introduction

Arbitrage-Free Trees

17.2 Applications and Uses of the Binomial Model

Stock Prices and Martingales

17.3 A Brief History of Options Pricing Models Option Pricing Pre-1973 Options Pricing, 1973 and After

17.4 An Example 17.5 The Assumptions 17.6 The Single-Period Model The No-Arbitrage Principle Building Binomial Trees

The Pricing Model The Hedge Ratio Risk-Neutral Valuation Actual versus Pseudo-probabilities

17.7 Summary 17.8 Appendix Proving the No-Arbitrage Argument The Probabilities and Risk Premium

17.9 Cases 17.10 Questions and Problems

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CHAPTER 17: SINGLE-PERIOD BINOMIAL MODEL

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17.1

Introduction

“Once in Hawaii I was taken to see a Buddhist temple,” wrote physicist Richard Feynman in The Meaning of It All: Thoughts of a Citizen Scientist. “In the temple a man said, . . . ‘To every man is given the key to the gates of heaven. The same key opens the gates of hell.’” Derivatives may be viewed as such a key. When the financial crisis of 2007 hit many countries, including the United States, numerous commentators, and even the lay public, blamed it on derivatives. Indeed, credit rating agencies failed to correctly rate complicated derivatives, and the losses on these derivatives brought down many financial institutions. Conversely, the growth of the derivatives market has been praised as improving economic welfare by shifting risks from those who fear it to those who profit from it. And derivatives help solve financial problems. They were even used by governments during the crisis to help resolve it. For example, the US Treasury started the Troubled Asset Relief Program during the financial crisis to commit up to “$700 billion to rescue the financial system,” notes the July (2009) Oversight Report of the Congressional Oversight Panel, “TARP Repayments, Including the Repurchase of Stock Warrants.” TARP purchased stock in the stressed banks with “troubled assets,” but in addition, they received ten-year warrants. Warrants are call options—derivatives! After the banks recovered, the US government sold these warrants back to the banks or to others at a “fair price,” determined with the help of the Black–Scholes–Merton (BSM) and binomial models. In doing so, they profited substantially. Note the use of the binomial model! This chapter develops the single-period binomial model with an eye toward understanding the major tenets of options pricing. We provide an illustrative example, give the model’s intuition, and state the necessary assumptions. To value the option, we construct a synthetic option with identical payoffs using the stock and a money market account. This takes us to martingale pricing and risk-neutral valuation, which lies at the heart of modern theory and which enables us to understand the more complex option pricing models (OPMs) that follow. Chapter 18 extends this chapter to a multiperiod setting and makes the binomial model practical and useful. The subsequent two chapters adopt a similar pattern: chapter 19 introduces the BSM model, and chapter 20 discusses the model’s practical use.

17.2

Applications and Uses of the Binomial Model

Options pricing bewildered and baffled academics until Fischer Black, Myron Scholes, and Robert Merton developed an analytic pricing model for European options in 1973. The mathematics underlying the model’s derivation is difficult. To more simply derive the model, as William Sharpe noted, one can use a binomial model and then take limits. Although the binomial model had been used as a teaching tool at the Massachusetts Institute of Technology (MIT) and other places before its

A BRIEF HISTORY OF OPTIONS PRICING MODELS

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publication, it was first printed in Sharpe’s classic textbook Investments. Subsequently, Cox, Ross, Rubinstein, Rendleman, Bartter, Jarrow, and Rudd developed popular versions of the binomial model. The model gets its name binomial (bi means “two”) from the assumed stock price evolution. From any point in time onward, over the next time step, the stock price can take only one of two possible values. This raises two immediate questions: (1) why do we need to make an assumption about stock price movements, and (2) is it reasonable to assume that stock prices can only take two values? The last chapter explicitly answered the first question. There we pushed as far as we could without imposing a model for the stock price evolution. Although we got many results, we could not obtain a pricing model. One needs to assume a model for the stock price evolution like the binomial (which we do in this and the next chapter) or a lognormal (chapter 19) to develop an exact OPM. As for the second question, do stocks really only take one of two values at each time step? No. But we need to start somewhere. We start with the simplest model to facilitate understanding and then add the necessary layers of complexity. The simple binomial OPM possesses some desirable features: ■

It approximates the BSM model. The binomial OPM can be repeated at each time step in a multiperiod tree. When the parameters are properly chosen (see Extension 18.1) and the model is run over many small time intervals, the multiperiod binomial OPM prices closely approximate BSM model prices.

■

It illustrates martingale pricing, the central idea of derivatives pricing. The binomial approach utilizes “martingale pricing,” a powerful technique for valuation. Once you understand this approach, you will find it easier to understand more complex OPMs.

■

It provides a useful numerical approximation technique. The binomial model is a versatile tool that gives numerical values to approximate solutions to exact pricing models (like the BSM model) or provides answers when no explicit analytical solutions exist (as in the case of an American put). Later chapters show how to use the binomial model to price interest rate derivatives. Even proprietary trading desks of Wall Street firms sometimes use sophisticated versions of the binomial pricing model.

17.3

A Brief History of Options Pricing Models

The history of OPMs has two phases: before and after 1973.

Option Pricing Pre-1973 It all started more than a century ago, when a young French mathematician, Louis Bachelier (1900), wrote a dissertation titled “The Theory of Speculation” that

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developed a prototype OPM. Despite some shortcomings, it was the forerunner of the BSM model. Bachelier was the first to model stock prices as a random walk. But fate was unkind, and he did not get the recognition he deserved during his lifetime. Within a decade after Bachelier’s death, his work resurfaced. It attracted the attention of Professor Paul Samuelson, the first American to win the Nobel Prize in Economics. During the 1960s, Samuelson and others worked on improving Bachelier’s model (see Table 17.1 for a summary of landmark achievements in this field).

TABLE 17.1: Some Milestones in the History of Option Pricing Models

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Year

Development

1900

Bachelier developed the first option pricing model (OPM).

1960s

Samuelson and other scholars improved Bachelier’s model.

1973

The Black–Scholes and Merton models were published.

1970s, 1995

Merton (1974, 1977) and Jarrow and Turnbull (1995) models for pricing derivatives with credit risk were introduced.

1978–82

The binomial model was presented in Sharpe (1978), Cox et al. (1979), Rendleman and Bartter (1979), and Jarrow and Rudd (1982).

1979 and 1981

Harrison and Kreps (1979) and Harrison and Pliska (1981) developed the martingale pricing methodology.

1987 and 1992

The Heath–Jarrow–Morton model (introduced in 1987, published in 1992), an interest rate OPM, was introduced.

1997

Merton and Scholes, the surviving co-originators of the BSM model, won the Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel.

Options Pricing, 1973 and After During the late 1960s, two young PhDs, Fischer Black and Myron Scholes, teamed up to work on options pricing. Around the same time, Robert Merton, a PhD student of Paul Samuelson, started working on the same problem. Combining their insights, 1973 saw the publication of the Black–Scholes–Merton model (Chapter 19 provides a more detailed history), which forms the foundation for today’s OPMs. Their work stands as a towering achievement not only in finance but also in the field of economics. After the watershed year of 1973, a race began to build better pricing models. Numerous OPMs resulted. These models priced different kinds of derivatives and relaxed the BSM model’s assumptions. The race continues to this day with unabated fury.

A BRIEF HISTORY OF OPTIONS PRICING MODELS

Why develop better OPMs? There are many reasons. The opening of options exchanges and the expansion of over-the-counter options trading created a need for better hedging. Brokers and dealers need OPMs to make effective marketsmarkets. The OPM extensions can be classified under five headings: 1. Improving the BSM model and using it for different underlyings. Soon after the publication of the BSM came a slew of models that improved the original. They included Black’s model to price commodity contracts. The stock price evolution was generalized, and market imperfections like short selling restrictions and transaction costs were included as well. 2. Including a term structure of random interest rates. A major drawback of the BSM model is its constant interest rate assumption. Early attempts at pricing interest rate options required estimating risk premia (e.g., Vasicek 1977; Brennan and Schwartz 1979) that made the models difficult to use. The removal of the need to estimate interest rate risk premia came in 1992 when the Heath–Jarrow–Morton (HJM) model was published. All arbitrage-free interest rate OPMs are special cases of the HJM model.1

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3. A rigorous mathematical foundation. Harrison, Kreps, Pliska, and others generalized the mathematics behind the BSM model. They developed martingale pricing, which soon became the most widely used technique for pricing derivatives. 4. Pricing derivatives with credit risk. Merton (1974, 1977) pioneered the pricing of derivatives with credit risk with the structural approach. Jarrow and Turnbull (1995) developed an alternative method that relaxes some restrictive assumptions inherent in Merton’s formulation. The Jarrow–Turnbull approach is called the reduced-form credit risk model. See chapter 26 for additional discussion. 5. Numerical methods. With more complex pricing models came the need to compute their values quickly and efficiently on computers. The binomial model was one of the first successful numerical methods (see Sharpe 1978; Cox et al. 1979; Rendleman and Bartter 1979; Jarrow and Rudd 1982). Others include numerical methods for solving partial differential equations and Monte Carlo simulation techniques. To understand options pricing theory, we begin with an example that illustrates the key ideas.

1

Baxter and Rennie (1996) state, “In the interest-rate setting, Heath–Jarrow–Morton is as seminal as Black–Scholes. By focusing on forward rates and especially by giving a careful stochastic treatment, they produced the most general (finite) Brownian interest-rate model possible. Other models may claim differently, but they are just HJM with different notations.”

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17.4

An Example

This section illustrates the binomial options pricing approach through an example (see Example 17.1).

EXAMPLE 17.1: European Call Pricing ■

Your Beloved Machine Inc.’s (YBM) stock price S is $100 today (time 0). After one year (time T), the stock price S(1) can either go up to $120.00 or down to $90.25. Let the actual probability of the stock going up q = 3/4; then that of YBM going down is (1 – q) = 1/4.

■

Consider a European call option with strike price K = $110 and maturity date time 1. The call’s payoff at time 1 is max[S(1) – 100, 0], that is, it is 10 = max(120 – 110, 0) if the stock goes up and 0 = max[90.25 – 110, 0] if the stock goes down.

■

Let a money market account (mma) cost $1 per unit at time 0. It earns a continuously compounded interest rate r = 0.05 per year. A dollar invested in the mma grows to $1.0513 (see Figure 17.1).

Synthetic call construction ■

Form a portfolio V with m shares of the stock and b units of the mma. Its value today is V (0) = m100 + b

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■

(17.1)

We want to choose (m, b) such that the portfolio’s payoffs at time 1, V (1), equal those of the traded call on the expiration date. This holds if the following two equations are satisfied: m120.00 + b1.0513 = 10

(17.2)

m90.25 + b1.0513 = 0

(17.3)

■

Solving these equations gives m = 0.3361 and b = –28.85, where a positive sign indicates a long and a negative sign a short. We call this portfolio the synthetic call.

■

Plugging the solutions into expression (17.1) gives V (0) ≡ m100 + b = $4.76. This is the cost of constructing the synthetic call at time 0. To avoid arbitrage, this must be today’s price for the traded call, c.

■

It is interesting to note that the option’s value was determined without using the actual up probability q = 3/4. This is an important observation. We will explain the reasons for this shortly.

■

Note that expression (17.1) also has a hedging interpretation. If you buy V (0) (a call option) and want to hedge the stock price risk, then sell m shares of the stock. This creates the hedged call portfolio V (0) – m100 = b, which is riskless because it’s equivalent to a position in the mma. We will also explain this hedging interpretation more completely later in this chapter.

Arbitrage Profits ■

What happens when the traded call’s price differs from the price of the synthetic call? Seize the opportunity and make arbitrage profits by buying low, selling high, or as the British would say, buying cheap, selling dear.

AN EXAMPLE

■

Suppose an errant trader quotes $7 for the traded call. As the traded call is more expensive, sell it and buy the synthetic call. Let’s do the accounting to see if it all works out. - Sell the overpriced traded call for $7 and purchase the underpriced synthetic call by buying 0.3361 shares for $33.61 and borrow $28.85 by shorting the mma at a net cost of $4.76. - Today’s cash flow is + 7 – 4.76 = $2.24. This is the immediate profit. - You have no liabilities in the future. If the stock rises at maturity, you owe $10 as the writer of the traded call. This is offset by the $10 that you get as the buyer of the synthetic call. Both calls expire worthless if the stock falls instead.

■

If a trader quotes $3 for the call, simply reverse your strategy: buy the traded call and simultaneously sell the more expensive synthetic call by short selling 0.3361 shares for $33.61 and lending $28.85 by buying the mma. You immediately make $1.76, and there is no future net exposure as the assets and liabilities perfectly match.

FIGURE 17.1: Single Period Binomial Example and Model Numerical Example Stock

Call

Today

After 1 year 3/4

Option starts

120.00

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100

Option matures

3/4

10

1/4

0

c

1/4 90.25 Call: strike price K = 110, maturity time 1 One-Period Model Stock

Call

Now

Maturity q

Time 0

US

Time 1 q

cU

(1 – q)

cD

c

S DS (1 – q) where U > D and 1 > q > 0

Call: strike price K, maturity time 1, payoff c(1) = max[S(1) – K, 0] Money Market Account (mma) 1 1.0513 1 + R er

t

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Put Pricing ■

Consider a European put with a strike price of K = $110 and a maturity of one year. As with the call, set up a portfolio to match the put’s time 1 value of $0 = max(110 – 120, 0) in the up state and $19.75 = max(110 – 90.25, 0) in the down state. This yields two equations in two unknowns (m, b): m120.00 + b1.0513 = 0 m90.25 + b1.0513 = 19.75

■

Solve the two equations to get the stock shares m = – 0.6639 and the number of mmas needed b = 75.78.

■

As the stock and put prices move in opposite directions, it makes intuitive sense that the replicating portfolio combines a short position in the stock and a long position in the mma.

■

The European put option’s price p is equal to the cost of construction at time 0, that is, V (0) ≡ m100 + b = − 0.6639 × 100 + 75.78 = $9.39

Put–Call Parity ■

Alternatively, you can derive the put’s value using put–call parity for European options:

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Put price = Call price + Present value of strike − Stock price = 4.76 + 110 × (1/1.0513) − 100 = $9.39 Any other put price will open up arbitrage opportunities.

17.5

The Assumptions

Assumptions are the key to understanding and using any model. Recall our discussion of the cost-of-carry model and its assumptions from Chapter 11. We need those same assumptions (A1 to A5) plus two more (A6 and A7). One is needed to describe the behavior of interest rates, and the other is needed to characterize the binomial stock price evolution. For emphasis as well as a better understanding of the model, we repeat our discussion of assumptions A1–A5. A1. No market frictions. We assume that trading involves no market frictions such as transaction costs (brokerage commissions, bid/ask spreads), margin requirements, short sale restrictions, and taxes (which may be levied on different securities at different rates). Moreover, assets are perfectly divisible. This helps us to create a benchmark model to which one can later add frictions. In addition, it’s a reasonable approximation for many traders such as institutions.

THE ASSUMPTIONS

Moreover, political changes, regulatory changes, and the information technology revolution have substantially reduced market frictions.

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A2. No credit risk. We assume that there is no credit risk. Also known as default risk or counterparty risk, this is the risk that the counterparty will fail to perform on an obligation. The absence of credit risk is a reasonable assumption for exchange-traded derivatives. Over-the-counter derivatives, however, may possess substantial credit risks, unless there are adequate collateral provisions. A3. Competitive and well-functioning markets. In a competitive market, traders’ purchases or sales have no impact on the market price; consequently, traders act as price takers. In a well-functioning market, price bubbles are absent. The competitive markets assumption is the workhorse of modern economics. Spot, forward, and futures markets for many agricultural commodities and precious metals behave like competitive markets. Pricing securities when counterparties have market power and traders can manipulate prices is a difficult exercise. One must take into account bargaining and strategic interactions, which are outside the scope of this book. A price bubble happens when an asset’s market price deviates from its intrinsic or fundamental value. The fundamental or intrinsic value is the price paid if, after purchase, you have to hold the asset forever. A difference between the market price and fundamental value only occurs if one believes that selling can generate a higher value than holding it forever. The difference is the price bubble. Recent research shows that bubbles can invalidate the methodology we employ for pricing options. Fortunately, however, one can show that there are no price bubbles possible in a discrete time binomial model (see Jarrow et al. 2010). A4. No intermediate cash flows. Many assets reward their holders with income or cash flows such as dividends for stocks and coupon income for bonds. For now, we assume that the underlying asset has no cash flows over the option’s life. The next chapter will show how to relax this assumption by introducing dividends. A5. No arbitrage opportunities. This is self-explanatory. These five assumptions are a repetition of the assumptions used in the cost-ofcarry model of chapter 11. Next we introduce two additional assumptions needed for deriving the binomial OPM. A6. No interest rate uncertainty. We assume that interest rates are constant across time. This assumption only works for short-lived options whose underlyings are not interest rates. It does not work well for long-dated options whose underlying assets’ prices are correlated with interest rate changes, like long-term options on foreign currencies. But long-dated options and interest rate derivatives are important in today’s markets, and their pricing is a prime challenge for researchers and practitioners. We relax this assumption in part IV of the book. Recall that we did not require this assumption for our cash-and-carry models. It wasn’t necessary because forwards are essentially valued in a single-period setting. However, we will soon be introducing multiperiod and continuous time models in

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which bond investments have reinvestment risks. To avoid this problem, we assume a constant interest rate. Assumption A6 will be maintained for all OPMs in part III of the book. A7. Binomial process. We assume that the stock trades in discrete time and that its price evolves according to a binomial process. For a current stock price, next period’s stock price will either get multiplied by an up factor U > 1 and take a higher value U × Stock price or it will be multiplied by a down factor D and take a lower value D × Stock price. We let q > 0 denote the actual probability of going up. We also assume U > D. Otherwise, we have mislabeled up and down. Assumption A7 will be with us in this and the next chapter. It will eventually be replaced by the assumption of “continuous trading” and the evolution of the stock price according to a “lognormal process” in the context of the BSM model.

17.6

The Single-Period Model

To develop the model, we proceed through a sequence of steps. First, we explain the no-arbitrage principle used in the example. Second, we show how to build a binomial tree that is arbitrage-free. Next comes the martingale representation of stock prices, which helps us to develop fancy entities called pseudo-probabilities. Then we obtain the call’s value by building on Example 17.1. Finally, we introduce an alternative perspective on options pricing called risk-neutral valuation. The next chapter will extend this to develop a multiperiod binomial OPM that approximates the BSM model.

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The No-Arbitrage Principle Let’s summarize the no-arbitrage principle used in the previous example to find the option’s value. ■

We considered a market in which both a stock and an option traded.

■

The stock took one of two possible values at the maturity date, and so did the option because the option’s payoff is completely determined by the stock’s ending value and the strike price.

■

We created a portfolio of the stock and mma to match the option’s payoffs in each of these time 1 states. This replicating portfolio is called the synthetic option.

■

By the law of one price—the no-arbitrage principle—the cost of constructing the synthetic option must equal the price of the traded option.

■

Why should they be the same? If the traded option and the synthetic option differ in price (see Figure 17.2), then you have two distinct ways of getting the same cash flows. Shrewd traders will make arbitrage profits by buying the cheaper option and selling the dearer option until their prices converge.

■

This pricing methodology has an interesting by-product. A portfolio that holds one of these options long and the other short should be entirely riskless. This kills two birds with one stone: pricing and hedging problems are solved in a single stroke!

THE SINGLE-PERIOD MODEL

FIGURE 17.2: The No Arbitrage Principle Portfolio A (Market traded option)

Portfolio value in up state

Must be equal without arbitrage Portfolio B (Synthetic option)

Portfolio value in down state

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Building Binomial Trees We need some notation to build a binomial tree. Today at time 0, the stock price is S(0) or S. At time 1, it can move up to S(1)UP or slide down to S(1)DOWN . Let’s write S(1)UP = SU and S(1)DOWN = SD, where the positive numbers U > D are called the up and down factors, respectively (Figure 17.3 shows how a stock price path looks in a single-period tree). This multiplicative binomial model has the advantage of ruling out negative stock prices. Moreover, in the multiperiod setting, it makes life easier by insuring that the tree’s branches recombine. To reduce clutter, we drop the dollar sign from the tree. The other traded security is a mma. It earns the risk-free rate in each period. A dollar gives (1 + R) = exp(rΔt) ≡ erΔt at the end of each period, where r is the continuously compounded yearly interest rate and Δt is the length of the time

FIGURE 17.3: Stock Price Evolution in a Single Period Setting Stock price Realized stock price path

US S DS

Possible path

0 Today

Time 1

Time

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CHAPTER 17: SINGLE-PERIOD BINOMIAL MODEL

period. In Figure 17.1, a single arrow is used to show the mma’s value because it earns a fixed return. Consider a European call option with strike price K and maturity time 1. Let c(0) or c denote its time 0 value. Given the call’s boundary condition, the call’s values at date 1 are given by max[0, S(1) – K]. We label this as c(1)UP ≡ cU when the stock price goes up and c(1)DOWN ≡ cD when it goes down (we drop some letters to reduce clutter). Figure 17.1 shows this model under the binomial tree example. Put values pU and pD could have been similarly introduced.

Arbitrage-Free Trees

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“House built on a weak foundation, will not stand oh no,” sang “King of Calypso” Harry Belafonte, who popularized Caribbean music. The same can happen with a model. Because we price options with a binomial tree using the no-arbitrage principle, the tree must be arbitrage-free for logical consistency. Otherwise, the whole approach collapses, like a house without a strong foundation. A simple condition Up factor > Dollar return > Down factor (U > [1 + R] > D) is both necessary and sufficient to rule out arbitrage profits in the binomial tree (see the appendix to this chapter for a proof). It states that neither the stock’s nor the mma’s returns dominate the other, a necessary condition for an economic equilibrium—and this makes intuitive sense. If one security is superior to the other, why invest in the inferior asset? This common sense arbitrage-free condition is trivial to impose and easy to understand. It must also hold in the multiperiod binomial tree, and it can be generalized. For example, even if U, (1 + R), and D depend on the node in the tree, the same condition must hold for all nodes. This will prove useful in Part IV of the book, when we price interest rate options.

Stock Prices and Martingales Before pricing options, let us study the no-arbitrage condition to see if we can obtain any additional insights. Suppose that U > (1 + R) > D holds in the stock price tree so that arbitrage is ruled out. Algebra tells us that U > (1 + R) > D holds if and only if we can find a unique number 𝜋 strictly between 0 and 1 such that (1 + R) = 𝜋U + (1 − 𝜋) D

(17.4)

(You can see this intuitively: as U is large and D is small, you can alter 𝜋 to create any number that lies between them.) Dividing both sides of Equation 17.4 by (1 + R) and multiplying by S gives S=

𝜋US + (1 − 𝜋) DS 1+R

(17.5)

This expression has a useful interpretation. Look at the stock tree in Figure 17.1. Suppose we say that the “probability” of the stock going up is 𝜋, and that of going down is (1 – 𝜋). Then, the stock’s expected payoff computed with these probabilities and discounted to the present by the riskless rate (1 + R) is today’s stock price! This

THE SINGLE-PERIOD MODEL

gives us a simple method for computing the stock’s present value that is consistent with no arbitrage. Given a stock price tree, it is easy to determine 𝜋. Indeed, solving Equation 17.4 gives (1 + R) − D U − (1 + R) 𝜋= and (1 − 𝜋) = (17.6) U−D U−D What have we done? Our no-arbitrage condition implies the existence of some numbers, 𝜋 and (1 – 𝜋), which we have interpreted as probabilities, and we have used them to compute the stock’s present value. However, it is important to emphasize that they are not the actual probabilities of the stock going up or down (which are q = 3/4 and [1 – q] = 1/4 in this case), so we call them pseudo-probabilities (or martingale probabilities or risk-neutral probabilities). The last two names will make sense shortly. Next we rewrite Equation 17.5 two more times. Why? Because after it is rewritten, it is easier to see how the binomial model relates to the BSM OPM studied in Chapter 19. We thus rewrite Equation 17.5 as (with T = 1)

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S = E𝜋 [S (T)] e−rT

(17.7a)

where E𝜋 (.) is shorthand for denoting that we are computing an expected value by using the probabilities 𝜋 and (1 – 𝜋), and we replaced (1 + R) with erT . This method of computing the stock’s present value as its expected payoff using the pseudo-probabilities and discounting it backward through time with the mma’s value (expression [17.7a]) is also known as martingale pricing. To see this interpretation, we need to rewrite this equation using the notation for a mma’s value. Let A and A(1) = er be the time 0 and time 1 value, respectively, of the mma. Then, Equation 17.7a is S (T) S = E𝜋 (17.7b) [ A (T) ] A This expression shows that S(t)/A(t) is a martingale under the pseudo-probabilities. What is a martingale? It’s a stochastic process X(t) whose time t value equals the expected value of X(T) at some later time T. Martingales are associated with “fair gambles” because what you have today is what you expect to have tomorrow. In finance, this sense of fairness gives us a pricing system with no arbitrage opportunities. Because of Equation 17.7b the pseudo-probabilities are often called the martingale probabilities.

The Pricing Model Inspired by the example, we create a synthetic call using a replicating portfolio V (0) [≡ mS + b] with m shares of the stock worth S per share and b units of the mma priced at a dollar each. At time 1, we want to find (m, b) to match the portfolio’s value to the traded call’s value in each state. This yields two equations in two unknowns: in the “up” state ∶

V(1)UP = mUS + b (1 + R) = cU

(17.8a)

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in the “down” state ∶

V(1)DOWN = mDS + b (1 + R) = cD

(17.8b)

where cU = max (US − K, 0) and cD = max (DS − K, 0)

(17.8c)

A solution exists and solving them gives m = (cU − cD ) / (US − DS) b = (UcD − DcU ) / [(U − D) (1 + R)]

(17.9a) (17.9b)

m is known as an option’s delta or the hedge ratio. We will soon show why it is the holy grail of options pricing. The cost of constructing this portfolio is V (0) ≡ mS + b cU − cD UcD − DcU S+ (U − D) S (U − D) (1 + R) (1 + R) − D U − (1 + R) 1 c + c = 1 + R [( U − D ) U ( U − D ) D ]

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=

(17.10)

Now to prevent arbitrage, the synthetic call’s cost of construction, V (0), must equal the traded call’s price, c. This is the option’s arbitrage-free price! As seen in Equation 17.10, this value depends only on (S, U, D, R, K), which we refer to somewhat flippantly as KRUDS. These are the current stock price (S), the up and down factors (U, D), the dollar return on the mma (1 + R), and the contract’s strike price (K). Missing from the call’s value are q and (1 – q), the actual probabilities of the stock going up or down. This omission is important and it occurs because, in the derivation of the option’s price, we use an exact replication argument. No matter which node of the tree the stock price moves, the synthetic call’s payoffs exactly duplicate the option’s payoffs. Hence we don’t care what the actual probability is of moving up or down. The omission of the actual probabilities from the option’s price is important because it implies that if two investors agree on the KRUDS, but they disagree on q, they will still agree on the call’s price. Of course, disagreement about the probability q might lead to disagreement about the stock price S when (U, D, R) are fixed. But if agreement about S occurs (which you can readily see for actively traded liquid stocks), then the call’s price is fixed by our no-arbitrage argument.

The Hedge Ratio Classical myths depict the Holy Grail as the cup used by Jesus Christ at the Last Supper. Believed to possess miraculous powers, it has been lost and sought for centuries. Indeed, the quest for the lost Holy Grail features prominently in the tales of the legendary knights of King Arthur’s court. Options pricing theory also has its holy grail, the hedge ratio, the number of shares of the stock to hold for each written call to form a perfect hedge. This hedge ratio has miraculous powers because with it one

THE SINGLE-PERIOD MODEL

can construct a synthetic call to obtain the arbitrage-free price of a traded option. Without it, our argument fails. The hedge ratio for the binomial model is given by Equation 17.9a: m=

cU − cD (U − D) S

To prove this, consider a portfolio consisting of the short call and m shares of the stock. The initial value of this portfolio is − c + mS The values of this portfolio at time 1 in the up and down states are: cU D − cD U (U − D) cU D − cD U − cD + mDS = (U − D) − cU + mUS =

Because these values are equal, the covered call is riskless—and we have found the holy grail! To reemphasize, the hedge ratio works perfectly if the world behaves according to our single-period model. Finding a hedge ratio that works for more complex and realistic OPMs is a much harder task. Our quest for the holy grail (of options pricing) continues in the coming chapters for these more complex OPMs.

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Risk-Neutral Valuation Another interesting insight can be obtained using some additional algebra on Equation 17.10. We can use the pseudo-probabilities to rewrite the call’s value in another form, which has a nice economic interpretation. Recall that Equation 17.6 gave 𝜋 = [(1 + R) – D]/(U – D) and (1 – 𝜋) = [U – (1 + R)]/(U – D). Using them, we can rewrite the call’s arbitrage-free value (17.10) as c = [πcU + (1 − π) cD ] / (1 + R)

(17.11)

This equation is an example of the risk-neutral valuation procedure. The option’s value is its expected payoff using the pseudo-probabilities and discounted using the mma’s rate. As the legendary baseball coach Yogi Berra said, “it’s like déjà vu, all over again!” The pseudo-probabilities 𝜋 and (1 – 𝜋) are back to give us a simple way of computing present values! This interpretation gives the pseudo-probabilities an alternative name: risk-neutral probabilities. Using Equation 17.8c, we can write (17.11) compactly (using T = 1) as c = E ᄡ {max [0, S (T) − K]} e−rT

(17.12)

which will be shown later to be similar to an expression used to express the BSM model in Chapter 19.

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EXAMPLE 17.2: Risk-Neutral Pricing of European Options in a Single-Period Binomial Model ■

Consider the data from Example 17.2. The model starts at date 0 (time 0) and ends at date 1 (time T).

■

Three securities trade: - YBM’s stock price S is $100 today. The up factor U is 1.2000 and the down factor D is 0.9025.3 - A dollar invested in the mma earns r = 5 percent per year continuously compounded and gives a dollar return of (1 + R) = erT = e0.05 × 1 = $1.0513 in one year. - As the strike price K is $110, the option prices at maturity are cU = max(120 – 110, 0) = 10.00 and pU = max(110 – 120, 0) = 0 when the stock goes up to $120.00 and cD = max(90.25 – 110, 0) = 0 and pD = max(110 – 90.25, 0) = 19.75 when the stock goes down to $90.25.

■

Equation 17.6 gives the pseudo-probability of an up movement 𝜋 = [(1 + R) – D]/(U – D) = 0.5001.

■

The risk-neutral pricing equation (17.9) gives today’s option prices: c = [𝜋cU + (1 − 𝜋) cD ] / (1 + R) = $4.76 p = [𝜋pU + (1 − 𝜋) pD ] / (1 + R) = $9.39

These are the same prices that we got in Example 17.2.

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3

We derive them by using 𝜎 = 0.142470 in Jarrow–Rudd specification (see Extension 18.1).

Actual versus Pseudo-probabilities We noted that the pseudo-probabilities (𝜋 and [1 – 𝜋]) for up and down movement in the stock prices, respectively, differ from the actual probabilities (q and [1 – q]). But how are they related? Because the two sets of probabilities denote positive fractions, we can link them by introducing two positive terms 𝜙U and 𝜙D that denote an adjustment for risk: 𝜋 = 𝜙U q

(17.13a)

(1 − 𝜋) = 𝜙D (1 − q)

(17.13b)

The appendix to this chapter shows why 𝜙U and 𝜙D are an adjustment for risk. We note that 𝜙U and 𝜙D are not independent. Knowing one, the other can be computed since 𝜙U q + 𝜙D (1 – q) = 1. The intuition is simple. To compute present values, it is well known that one can take expected future values (using the actual probabilities) and discount to the present using a risk-adjusted rate. The larger the risk, the larger is the risk-adjusted discount rate. However, if one uses the pseudo-probabilities instead to compute a present value as in the risk-neutral valuation Equation 17.11, then the discount rate becomes the riskless rate. Because risk-neutral valuation does not adjust the discount rate for risk,

SUMMARY

to get the same present value, the adjustment for risk must occur in the use of the pseudo-probabilities (as distinct from the actual probabilities). This is indeed the case. Building on this insight, the appendix also discusses how these two sets of probabilities and the risk premium can be estimated using market data. We note that each set of probabilities has its own use: the actual probabilities contribute (in conjunction with historical data) to the estimation of the volatility, while the pseudo-probabilities (which we manufacture by using the risk-free interest rate, the length of a time interval, and the volatility; see Extension 18.1) are used for computing option prices. These actual and pseudo-probabilities will stay with us in the chapters that follow.

17.7

Summary

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1. In 1973, Fischer Black, Myron Scholes, and Robert Merton revolutionized the field of derivatives pricing by publishing the BSM model. Since 1973, options pricing developed along four, somewhat interlinked directions: (1) improving the BSM model and using it for different underlyings, (2) developing models that allow for random interest rates, (3) putting derivatives pricing on a rigorous mathematical foundation, and (4) using computational methods, including the binomial framework. 2. The binomial model has several advantages: (1) it is a great teaching aid that gives a simple, intuitive introduction to options pricing, (2) it can be repeated to build a multiperiod tree, which gives prices that closely approximate the BSM model, (3) it utilizes martingale pricing, the paramount technique of derivatives valuation, and (4) it is a versatile model that prices standard European and American options and other derivatives. 3. The binomial OPM makes some key assumptions: no market frictions, no credit risk, competitive and well-functioning markets, no interest rate uncertainty, no intermediate cash flows, and no arbitrage. In addition, it assumes that the underlying trades at discrete time intervals and follows a binomial process. 4. The OPM constructs a synthetic option that replicates a traded option’s payoff. Next, if no arbitrage opportunities exist, the two must have the same value. An option’s arbitrage-free price is the cost of constructing the synthetic option. 5. The binomial OPM is set up as follows: - Three securities trade: a stock, a riskless mma, and a European (call or put) option. - The stock price goes up and down by the factors of U or D, respectively. - A dollar invested in the mma gives a dollar return of (1 + R) after one period. - As the strike price is given, the option’s payoffs at maturity are determined by the stock prices on the tree. Solve for the option price by the formula Option price = [𝜋 × (Option payoff)UP + (1 − 𝜋) × (Option payoff)DOWN ] / (1 + R)

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where 𝜋 = [(1 + R) – D]/(U – D) is the pseudo-probability of an up movement, and option payoffs “up” and “down” are considered at the up and down nodes. - The method of computing the expected payoff with pseudo-probabilities and discounting with the mma’s value is known as risk-neutral valuation.

17.8

Appendix

Proving the No-Arbitrage Argument The no-arbitrage condition U > (1 + R) > D proves useful in many contexts. 1. Single-period binomial model: U > (1 + R) > D (“Bounded Dollar Return”): This is a necessary and sufficient condition for no arbitrage in a single-period tree. This is the result proven in this extension. 2. Multiperiod binomial model: This result must hold at every node in the multiperiod binomial tree. 3. Binomial interest rate derivatives pricing model: A generalized version of this result is required for pricing interest rate derivatives in a binomial framework: if U, (1 + R), and D depend on the node in the tree, then the bounded dollar return condition must hold for all nodes (see Jarrow, 2002b).

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4. Pseudo-probabilities: We will soon see that this condition is intimately connected with the existence of pseudo-probabilities (or equivalent martingale probabilities or risk-neutral probabilities) in binomial models. Thus the condition that we establish is linked to the core ideas at the heart of options pricing.

RESULT 1 No-Arbitrage Condition The condition U > (1 + R) > D is equivalent to no arbitrage.

PROOF OF PART I: NO ARBITRAGE IMPLIES U > (1 + R) > D. Assume the condition U > (1 + R) > D fails. We show that this gives arbitrage profits. A violation of U > (1 + R) > D can happen in one of two ways—either (1 + R) is to the left side of U > D (i.e., [1 + R] ≥ U > D) or to its right (i.e., U > D ≥ [1 + R]). ■

Suppose (1 + R) ≥ U > D. This makes the mma a better investment than the stock, irrespective of the stock’s going up or down. Suppose the stock price is $100 today. Short sell the stock and invest the 100 proceeds in the mma. At the end of the period, your arbitrage profit is 100[(1 + R) – U] ≥ 0 if the stock goes up or 100[(1 + R) – D] > 0 if the stock goes down. This is an arbitrage strategy.

APPENDIX

■

Suppose U > D ≥ (1 + R). This makes the mma an inferior investment to the stock. Borrow at the risk-free rate by short selling the mma, invest the proceeds in the stock, and collect arbitrage profits.

PROOF OF PART II: U > (1 + R ) > D IMPLIES NO ARBITRAGE. Recall from chapter 6 that arbitrage can be expressed as a costless strategy that nonetheless pays something in the future. With two assets, a stock, and a mma, how would you obtain a zero-investment portfolio? Obviously, you cannot buy or sell both assets simultaneously (how would you finance such trades?). You must buy one asset and short the other. However, you cannot make arbitrage profits if U > (1 + R) > D holds. Why? Because in one of the up or down nodes, the payoff to the portfolio will be negative. To see this, suppose that you buy the stock and short the mma obtaining a zero initial investment portfolio. The payoff at time 1 if you go up is U – (1 + R) > 0. The payoff if you go down at time 1 is D – (1 + R) < 0. This portfolio is not an arbitrage opportunity. Conversely, if you short the stock and long the mma, the zero initial investment portfolio has a negative payoff in the up node and a positive payoff in the down node. This zero investment portfolio is also not an arbitrage opportunity.

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The Probabilities and Risk Premium RELATING ACTUAL AND PSEUDO-PROBABILITIES Let us further explore the concepts of the actual probabilities (q, 1 – q) and the pseudo-probabilities (𝜋, 1 – 𝜋) in the context of our single-period model and compute the risk premium that connects them. First, the probabilities must agree on zero-probability events. If one of the actual probabilities is zero, the stock price tree will fail to branch out and the stock becomes a riskless asset. The condition q > 0 if and only if 𝜋 > 0 prevents this collapse. Next, recall that we linked the pseudo-and actual probabilities by Equations 17.13a and 17.13b: 𝜋 = 𝜙U q (1a) (1 − 𝜋) = 𝜙D (1 − q)

(1b)

where 𝜙U and 𝜙D are positive numbers that adjust for risk. We note that 𝜙U and 𝜙D are not independent. Knowing one, the other can be computed since 𝜙U q + 𝜙D (1 – q) = 1. You can write them as two values taken by the random variable 𝜙.̃ To summarize, today’s stock price is S, and the parameters take on the following values after one period: S (1) =

US is associated with q > 0, 𝜋 > 0, and 𝜙̃ = 𝜙U {DS corresponds to (1 − q) > 0, (1 − 𝜋) > 0, and 𝜙̃ = 𝜙D

It follows from this definition that the expectation of the random variable 𝜙̃ computed using the actual probabilities is given by E (𝜙̃ ) ≡ 𝜙U q + 𝜙D (1 − q) = 𝜋 + (1 − 𝜋) = 1

(2)

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THE RISK PREMIUM Equation 17.5 allows us to write the present value of the stock price at time 0 as the discounted expected payoffs using the pseudoprobabilities: 𝜋US + (1 − 𝜋) DS (3) S= 1+R Using (1a) and (1b), we rewrite this as S=

q (𝜙U US) + (1 − q) (𝜙D DS) 1+R

(4)

Notice that we are adjusting for risk via the cash flows in the numerator. This contrasts with the traditional way of computing a present value. The traditional method is to adjust the denominator. You may have taken courses in capital budgeting or project evaluation, in which you learned to compute the net present value by discounting risky cash flows, positive or negative, with a risk-adjusted discount rate. Interestingly, 𝜋 and q are related by a risk premium. Using algebra and statistics, we show that the risk premium is given by cov(−𝜙,̃ S(1)/S ). To establish this, first, note from Equation 17.17 that we have S=

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or 1 + R =

̃ (1)) E (𝜙S 1+R ̃ (1)) E (𝜙S S

(5)

where E(.) denotes the expectation computed by using the actual probabilities q and (1 – q). A familiar result in statistics (for two random variables x and y, cov[x, y] = E[xy] – E[x]E[y]) gives ̃ (1)) − E (𝜙̃ ) E (S (1)) cov (𝜙,̃ S (1)) = E (𝜙S

(6)

Using (2) and transferring the last expression to the left-side, we get ̃ (1)) cov (𝜙,̃ S (1)) + E (S (1)) = E (𝜙S

(7)

We can write (5) using (7) and another result from statistics [cov(ax, y) = acov(x, y)] as

1+R=

̃ E (𝜙S(1)) S (1) E (S (1)) = cov 𝜙,̃ + ( ) S S S

(8)

APPENDIX

Rearrangement of Equation 8 gives E (S (1)) S (1) S (1) = 1 + R − cov 𝜙,̃ = 1 + R + cov −𝜙,̃ ( ( S S ) S ) E (S (1)) or S = S (1) 1 + R + cov −𝜙,̃ [ ( S )]

(9)

We rewrite Equation (9) as the following result.

RESULT 2 The Risk Premium Linking Actual and Pseudo-probabilities In a single-period model, the actual and pseudo-probabilities are linked by a risk premium cov(−𝜙,̃ S(1) / S), which is given in the following expression:

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S=

q (US) + (1 − q) DS S (1) 1 + R + cov − 𝜙̃ , ( S )

(10)

where cov( , ) is the covariance operator.

Thus the risk premium is cov(−𝜙,̃ S(1)/S). As shown, the risk premium depends explicitly on the pseudo-probabilities, proving they give an adjustment for risk. The risk premium can be explicitly derived in an equilibrium model. Equilibrium models (such as the capital asset pricing model) require the equality of demand and supply, the presence of utility functions and endowments, which determine risk premium (defined as the expected risky return less the risk-free rate). By contrast, arbitrage models (like our options pricing models) do not need such features and use only the no-arbitrage condition to derive the results. Notice that the difference in the two approaches to computing present values is that we only need to use arbitrage pricing for expression (3), but for expression (10), we need to compute both 𝜙U and 𝜙D , which requires an equilibrium model.

ESTIMATING PROBABILITIES How would you compute the probabilities? As noted before, the computation of the pseudo-probabilities (𝜋, 1 – 𝜋) is given in Equation 17.6. The actual probabilities can be computed using historical data and

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statistics for a single-period binomial distribution. First, using historical data, estimate the expected return Ê (return); then set it equal to the theory’s expected return, Ê(return) = qU + (1 – q)D, and solve for q. Remember that this is a toy model, and although we have shown how to solve for q, the estimation issues become more involved in the case of more complex models—but they use the same ideas as shown here!

17.9

Cases

Leland O’Brien Rubinstein Associates Inc.: Portfolio Insurance (Harvard Busi-

ness School Case 294061-PDF-ENG). The case studies the rise and fall of Leland O’Brien Rubinstein Associates’ portfolio insurance selling business. Leland O’Brien Rubinstein Associates Inc.: SuperTrust (Harvard Business

School Case 294050-PDF-ENG). The case examines Leland O’Brien Rubinstein Associates’ attempts to rebuild itself after the 1987 stock market crash by creating new products to meet the unsatisfied needs of equity investors. Tata Steel Limited: Convertible Alternative Reference Securities (B) (Richard

Ivey School of Business Foundation Case 910N32-PDF-ENG, Harvard Business Publishing). The case considers valuation of a convertible bond offering by a global top ten steel producer.

17.10 Questions and Problems Copyright © 2018. World Scientific Publishing Company Pte. Limited. All rights reserved.

17.1. Why is the binomial model a useful technique for approximating options

prices from the Black–Scholes–Merton model? Describe some applications and uses of this model. 17.2. How is the no-arbitrage principle used in the binomial model to find options

prices? 17.3. In the binomial options pricing model, it is assumed that the stock price

follows a binomial process. a. Is this a reasonable description of the actual stock price process? b. If not, why should one study this model? 17.4. a. In the binomial options pricing model, what assumptions are made about

dividends and interest rates? b. In the binomial stock price tree, what restrictions are needed on the up

and down factors, relative to the risk-free rate, to avoid arbitrage? Explain. 17.5. In the binomial options pricing model, what is the hedge ratio? 17.6. a. In the binomial options pricing model, what are the pseudo-probabilities? b. In the binomial options pricing model, what does risk-neutral valuation

mean? Explain.

QUESTIONS AND PROBLEMS

17.7. When pricing an option using risk-neutral valuation, one is assuming that all

investors are risk neutral. Hence, if one believes that investors are risk averse, risk-neutral valuation cannot be used. True or false? Explain your answer.

FIGURE 17.4: Today

After 1 year 120.773898

100 89.47150422 Call: strike price K = 105, option matures after 1 year. Money Market Account (mma)

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1

1.051271096

Note: The up factor U is US/S = 120.773898/100 = 1.2077. The down factor D is DS/S = 89.471504/100 = 0.8947 Dollar return 1 + R = 1.0513. We report numbers to four places and the final answer to two places after the decimal point although calculations consider ten places after the decimal point. The next eight questions use the above data. 17.8. a. Given the preceding data, set up a perfect hedge and compute the call

option’s value. b. What is the hedge ratio? What does it signify? 17.9. a. What are the pseudo-probabilities? b. Show that the stock price is its discounted expected value using these

pseudo-probabilities. 17.10. Compute the call option’s value using risk-neutral valuation. 17.11. Suppose the market price of the call option is $6. Is there an arbitrage

opportunity? Show how you can take advantage of this price to make arbitrage profits. 17.12. a. Set up a perfect hedge and compute the put option’s value. Let the put

have the same strike price K = $105. b. What is the hedge ratio? What does it signify?

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17.13. Compute the put option’s value using risk-neutral valuation. 17.14. Using the data from questions 17.8 and 17.11, verify that put–call parity holds. 17.15. For the data in question 17.13, suppose the market price of the put option is

$3. Is there an arbitrage opportunity? Explain how one would trade to exploit this arbitrage. The next four questions are based on the following data for a single-period binomial model: ■

A stock’s price S is $50. After six months, it either goes up by the factor U = 1.22095341 or it goes down by the factor D = 0.79881010.

■

Options mature after T = 0.5 year and have strike price K = $45.

■

A dollar invested in the money market account earns continuously compounded risk-free interest at 2 percent per year.

17.16. Compute the call’s value. 17.17. Suppose the market price of the call option is $5. Is there an arbitrage

opportunity? Explain how one would trade to exploit this arbitrage. 17.18. Consider the following exotic option whose payoff at expiration is given

by the stock price squared less a strike price if it has a positive value, zero otherwise:

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max [S(1)2 − K, 0] Assuming that the strike price K is $2,500, determine the value of this exotic option under the assumption of no-arbitrage. If the market price of the call is $600, how would you trade to exploit this arbitrage opportunity? 17.19. Consider the following exotic option whose payoff at maturity is given by

the square root of the stock price less the strike price if it has a positive value, zero otherwise: max [√S (1) − K, 0] Assuming that the strike price K is $7, determine the value of this exotic option under the assumption of no-arbitrage. If the market price of the call is $0.10, how would you trade to exploit this arbitrage opportunity? 17.20. (Microsoft Excel) Implied Volatility

Consider the following data for computing option prices given to you by your professor.

QUESTIONS AND PROBLEMS

FIGURE 17.5: Stock Today

After 1 year

120.77389800 100 89.47150422

Call: strike price K = 105, option matures after 1 year. Money Market Account (mma)

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1

1.051271096

You want to know how this data was generated. The professor, when asked, apologizes and says, “I used a Jarrow-Rudd approximation but I lost the data. You are an Excel expert—why don’t you use ‘Goal Seek’ and determine what the volatility is?”

391

18 Multiperiod Binomial Model Copyright © 2018. World Scientific Publishing Company Pte. Limited. All rights reserved.

18.1 Introduction

Recasting the Two-Period Example in the Multiperiod Framework

18.2 Toward a Multiperiod Binomial Option Pricing Model

The n-Period Binomial Option Pricing Model

The Stock Price Evolution

EXTENSION 18.1 Linking the Binomial Model to the Black–Scholes–Merton Model

Binomial Option Price Data The Stock Price Tree

18.3 A Two-Period Binomial Model 18.5 Extending the Binomial Model Backward Induction

Known Dividends

Option Pricing via Synthetic Construction (Method 1)

Valuing American Options

Repeat, Repeat: Risk-Neutral Pricing (Method 2)

18.6 Spreadsheet Applications A Two-Period Binomial Example

One-Step Valuation: Prelude to the Multiperiod Model (Method 3) 18.7

18.4 The Multiperiod Binomial Option Pricing Model Binomial Coefficients and Pseudo-probabilities

A Sixteen-Period Example

Summary

18.8 Cases 18.9 Questions and Problems

TOWARD A MULTIPERIOD BINOMIAL OPTION PRICING MODEL

18.1

Introduction

Not too many years back, a large US consulting company valued some options for a client. Two groups within the company were given the job. One of them used the celebrated Black–Scholes–Merton (BSM) option pricing model (OPM), which gives a neat analytical solution. This team (involving one of us) argued that the BSM model is useful because a formula enables the computation of changes in the option price in response to changes in the stock price (later, you will see how this helps traders to hedge and manage an options portfolio). The rival group used a multiperiod binomial OPM with thirty time periods and thirty-one branches in an Excel spreadsheet. They argued that the binomial model is easier for clients to understand and that it gives prices close to the BSM values. Both claims are correct, yet ease of understanding carried the day. This chapter extends the binomial model to a multiperiod setting and shows how to set the parameters to get prices that closely approximate the BSM. First, we develop a two-period binomial OPM repeating concepts and methods used earlier. Riskneutral valuation, when generalized, takes us to the multiperiod model. We also show the versatility of this numerical approximation technique when it is used to price American options and for including dividends. Finally, we discuss setting up the binomial model in a spreadsheet program like Microsoft Excel.

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18.2

Toward a Multiperiod Binomial Option Pricing Model

To construct the multiperiod binomial OPM, we first need to understand the assumed stock price process.

The Stock Price Evolution Figure 18.1 shows how a stock price path looks under the binomial assumption. The first figure shows the path in the single-period binomial model, which is the model from chapter 17. It’s a line parallel to the horizontal axis. Here it corresponds to the value US. Alternatively, it could have taken the value DS. In looking at this graph, one can easily reject the single-period model as a good approximation over, say, a one-year horizon when compared to the actual stock price path given in the next figure. The next figure also shows the stock price path in a multiperiod binomial setting. You can still reject this evolution just by observation, but what happens if these periods shrink and become really small? We have superimposed a realistic looking stock price path on this graph. When the time steps become small, the multiperiod binomial price process approximates this actual price path quite well! Here the multiperiod binomial model may not be so easily rejected. In fact, choosing the actual probability of going up q, the up factor U, and the down factor D appropriately, the multiperiod binomial evolution approaches the lognormal distribution that we use in the next chapter to obtain the BSM model.

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FIGURE 18.1: The Stock Price Evolution in a Single Period and Multiperiod Setting Single Period Stock price Realized stock price path

US S DS

Possible path

0 Today

After 1 year

Time

Multiple Periods Stock price Actual stock path Stock price path (model)

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U2S US S DS D2S

0 Now

1 year

Time

Binomial Option Price Data Our examples will use the binomial option pricing data (BOP data!) for Your Beloved Machines Corp. (YBM), which are the same data used in Example 17.1 in the last chapter. ■

YBM’s current stock price S is $100.

■

We consider a European call on YBM with strike price K = $110 and maturity time T = 1 year.

■

The model starts today (time 0) and ends after one year at time T = 1.

■

The stock price return volatility 𝜎 is 0.142470.

■

The number of periods n varies; it can be 1, 2, 3, or more. If n is 2, we divide the time to maturity T into two periods of length Δt = T/n = 0.5 year each. If n is 3, then Δt = 1/3 = 0.3333 year, and so on.

TOWARD A MULTIPERIOD BINOMIAL OPTION PRICING MODEL

■

We let the continuously compounded risk-free interest rate r be 5 percent per year. In the one-period model, the dollar return 1 + R = e0.05 = $1.0513 after one year. In the two-period model, the dollar return 1 + R = erΔt = e0.05×0.5 = $1.0253 after one period and (1 + R)2 = 1.02532 = $1.0513 after two periods—at year’s end. This is a benefit of continuous compounding. No matter how you slice and dice the time intervals, your investment still grows by the same amount over a year. Recall that all of our computations are performed to 16 decimal place accuracy, although for clarity in exposition, we report rounded numbers up to either 2 or 4 decimal places.

The Stock Price Tree The next step is to price options in a two-period binomial model. Example 18.1 presents the stock price tree and the money market account used for options pricing.

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EXAMPLE 18.1: A Stock Price Tree and Option Prices at Maturity ■

As before, a stock, a money market account, and a European call option trade. We use the BOP data. The model has three dates: today is the starting date (time 0), six months later is date 1, and the maturity is date 2 (time T = 1 year), when the call expires. As the number of periods n is 2, there are two time periods of length Δt = T/n = 0.5 years each.

■

The asset price evolution is shown in Figure 18.2. The dollar return is 1 + R = $1.0253 after one period. YBM’s stock price evolves according to a multiplicative binomial model. From any node in the binomial tree, the next period’s stock price is obtained by either multiplying the current stock price by the up factor U or by the down factor D. We compute these factors using specification 1 in Table 18.1 (see Extension 18.1): U = 1.1282 and D = 0.9224. - Starting from a price of S = $100 at date 0, the stock can either go up to US = 1.1282 × 100 = $112.82 or down to DS = $92.24 at the end of the first period. - If the stock price is US at date 1, then it either goes up to U(US) or down to D(US) at date 2. If it has a value of DS at date 1, then it subsequently goes up to U(DS) or down to D(DS). Hence U (US) = 1.1282 × 112.82 = $127.29 D (US) = 0.9224 × 112.82 = $104.07 = U (DS) D (DS) = 0.9224 × 92.24 = $85.08 - Notice that the “up and then down” dotted path combines with the “down and then up” dashed route because D(US) = U(DS). The branches combine because we use a multiplicative model. This feature is very useful when there are many periods. For example, if the branches recombine, a twoperiod model would have three instead of four separate nodes. However, in a 10-period model, there would be 10 + 1 = 11 separate nodes in a recombining tree but 210 = 1,024 separate nodes if the branches do not recombine. - A generalization of the result (U > 1 + R > D) of chapter 17 is essential to make the tree arbitragefree. We assume that this condition holds at every node in the tree.

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- The actual probability of the stock going up is q = 3/4 and that of YBM going down is (1 – q) = 1/4. Multiplying the probabilities together gives the probability of two up movements as q2 = 9/16, an up and a down movement as 2q(1 – q) = 3/8, and two down movements as (1 – q)2 = 1/16. ■

You can easily compute the options prices at date 2. - The call’s value at maturity is given by max[0, S(2) – K]. Consider the topmost node in Figure 18.2. The stock price there is U 2 S. Thus when the stock goes “up and then up,” the call value at time 2 is c(2)UP,UP ≡ cUU = max(U2 S – K, 0). Note that “UU” appears as a subscript in the call price. Also note that we drop “(2)” to reduce clutter. The call has a strike price K = $110. Hence cUU = max(127.29 – 110, 0) = 17.29. The call finishes out-of-the-money at the other two nodes on date 2, so c(2)UP,DOWN ≡ cUD = 0 and c(2)DOWN,DOWN ≡ cDD = 0. - The put’s value at date 2 is max[K – S(2), 0]. Using this formula, pUU = $0, pUD = $5.93, and pDD = $24.92.

18.3

A Two-Period Binomial Model

We will utilize the previous data to compute European options prices using a technique called backward induction.

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Backward Induction In Sir Arthur Conan Doyle’s first novel A Study in Scarlet (1887), Sherlock Holmes made a classic comment about his crime detection technique: “In solving a problem of this sort, the grand thing is to be able to reason backward. That is a very useful accomplishment, and a very easy one, but people do not practise it much.” Interestingly the approach used by Sherlock Holmes is called backward induction, a mathematical technique where one starts at the end and then solves a problem by working backward through time. We use it to solve the multiperiod binomial option pricing. The steps are as follows: ■

Create the binomial stock price tree by repeatedly attaching the single-period structure to the end of each node until the tree reaches the option’s maturity date.

■

Starting at the last date in the tree, given the strike price, you can easily compute the option’s values on the expiration date for all possible final nodes.

■

At the end of the last period, consider the option values ordered from the top node to the bottom node. Select the top two option values and apply the risk-neutral pricing formula to get the topmost option price at the previous date. Next, move down one node at the final time, and repeat this computation for the next two option values. Continue moving down the final nodes of the tree in this fashion, until the bottommost node is reached. When reached, this process generates all the option prices at the beginning of the last period from the top to the bottom of the tree.

A TWO-PERIOD BINOMIAL MODEL

FIGURE 18.2: Two-Period Binomial Model for Pricing a European Option Numerical Example Stock Today

Stock 6 months

Stock 1 year

Call 1 year

Put 1 year

Pseudo Prob Actual Prob 1 year 1 year

127.29

17.29

0

0.25

9/16

104.07

0

5.93

0.50

6/16

85.08

0

24.92

0.25

1/16

Stock Date 2

Call Date 2

Put Date 2

Pseudo Prob Actual Prob Date 2 Date 2

U2S

cUU

pUU

π2

q2

UDS

cUD

pUD

2π (1 – π)

2q (1 – q)

D2S

cDD

pDD

(1 – π)2

(1 – q)2

112.82 100.00 92.24

Two-Period Model Stock Date 0

Stock Date 1 π π

S (1 – π) (1 – q)

q

US (1 – π) q DS (1 – q)

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Money Market Account (mma)

■

Date 0

Date 1

Date 2

1

1.0253 1

1.0513 1.0253

Today

6 months

1 year

1

1+R

(1 + R)2

Step back one time step (toward date 0), and compute the option values at the beginning of the current period, using the procedure just described. Continue doing this period by period, working backward through time. Eventually you come to today and then the process ends. The solution is today’s option price.

Option Pricing via Synthetic Construction (Method 1) This section uses data from Example 18.1 to demonstrate how to price options in a two-period binomial framework. We begin by setting up a dynamic hedge using the stock and a money market account to replicate the call’s value. Dynamic hedging is the extension of our synthetic construction to a multiperiod setting.

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We use a dynamic, self-financing trading strategy to construct a portfolio that replicates the traded option’s payoffs. This portfolio is dynamic because it changes over the multiple periods, and it is self-financing because there are no net cash flows at intermediate dates. When no arbitrage is allowed, the cost of constructing this synthetic option must equal the price of the traded option.

EXAMPLE 18.2: Pricing a Call via Synthetic Construction ■

When solving the model, like Sherlock Holmes, we have to work backward from the end. Inspired by the example, we create at date 1 (one date before the last date) a portfolio V (1) ≡ m(1)S(1) + b(1) with m(1) shares of the stock worth S(1) and b(1) units of the money market account priced $1 each. This portfolio is constructed to replicate the option’s values at date 2. It is the synthetic call option. Its cost of construction must equal the traded call value at date 1, or else arbitrage opportunities arise.

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We have to do the replication exercise twice: once when the stock goes up to US and once more when it goes down to DS. In each case, we have to solve two equations in two unknowns to find the share holdings m(1), b(1).

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We use the same data as in Example 18.1 (see Figure 18.2 for a summary of this information) to illustrate this method. To reduce clutter, we use U and D in the subscripts instead of UP and DOWN, for example, V (2)UP,UP is expressed as V (2)UU ⋅ At date 2, when the stock is at US (= 112.82) on date 1, In the “up, up” state, V (2)UU = m (1) U2 S + b (1) (1 + R) = cUU

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In the “up, down” state,

V (2)UD = m (1) UDS + b (1) (1 + R) = cUD

Solving these equations, we get m(1)U = (cUU – cUD ) / (U2 S – UDS) = 17.29/ (127.29 – 104.07) = 0.7445 b(1)U = (UcUD – DcUU ) / [(U – D) (1 + R)] = – (0.9224 × 17.29) / [(1.1282 – 0.9224) 1.0253] = –75.57

(18.1)

Thus the cost of constructing the synthetic call at date 1 in the up state is V (1)U = m(1)U US + b(1) = 8.4345 = $8.43. To rule out arbitrage, this must equal the traded call’s price cU on date 1 in the up state. At date 2, when the stock is at DS (= 92.2363) on date 1, either solve the two equations to find that the call is worthless at date 1, or infer that if the call is sure to be worthless at date 2, then it must also be worthless at date 1. So when the stock goes down to 92.24, m(1)D = 0 and b(1)D = 0, giving the synthetic option a value of cD = 0 at date 1 in the down state. ■

Finally, let us now move back to date 0. Construct a portfolio V (0) ≡ m(0)S(0) + b(0) with m(0) shares of the stock worth S(0) and b(0) units of a money market account worth $1 each. This portfolio’s share holdings are constructed to match the synthetic call prices cU and cD at date 1, giving two equations in two unknowns. At date 1,

A TWO-PERIOD BINOMIAL MODEL

In the “up” state,

V (1)U = m (0) 112.82 + b (0) 1.0253 = 8.43 = cU

In the “down” state,

V (1)D = m (0) 92.24 + b (0) 1.0253 = 0 = cD

Solving these equations, we get m(0) = 0.4097 and b(0) = – 36.8536, so the cost of constructing the synthetic call at date 0 is V (0) = m (0) S (0) + b (0) = 4.1135 = $4.11 This must equal c(0) ≡ c, the traded call’s price today. Any other quoted price will create getrich opportunities for vigilant arbitrageurs. The numbers from this exercise are summarized in Figure 18.3.

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This simple example illustrates several important features of the multiperiod binomial OPM: ■

Prices differ from the single-period model prices. Despite using the same BOP data, the single- and two-period models give different options prices: $4.76 and $4.11, respectively. This is because the two trees aren’t identical. Compare the figures to notice that the stock and the option prices have different distributions at the maturity date.

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Self-financing. We show next that the dynamic trading strategy is self-financing. Initiate the trading strategy at date 0 by buying 0.4097 units of the stock priced at $100 and finance this by selling 36.8536 units of the money market account worth $1 each. You still need to invest $4.11, which is the call price. Suppose the stock goes up to $112.82 after six months. Then the portfolio’s value is $8.43. Liquidate this portfolio and raise funds by selling 75.57 units of the money market account for $1 each. Use the total proceeds of $84 (= $8.43 + $75.57) to buy 0.7445 units of the stock. Obviously, this transaction is self-financing. After the up node, suppose the stock goes up again to $127.29 at the end of the year. Then the portfolio’s value is (0.7445 × 127.29 – 75.57 × 1.0253) = $17.29. If the stock goes down from there to $104.07, then the portfolio value is (0.7445 × 104.07 – 75.57 × 1.0513) = 0. These payoffs are identical to the call values at the corresponding nodes in the tree at date 2. Suppose the stock goes down to $92.24 at date 1 instead. Then the call becomes worthless. No trades are necessary. The call has zero value thereafter.

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Dynamic market completion. Just two assets, a stock and a money market account, replicate the three option values at maturity. Note that we have dynamically completed the market, where the term complete refers to the portfolio’s ability to match all possible option values at maturity using a dynamic trading strategy.

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A hedging interpretation. Notice the holy grail, the hedge ratios m(0), m(1)U , and m(1)D in the model. They tell us how to rebalance our portfolios to synthetically construct the option values in the nodes that follow.

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FIGURE 18.3: The Two-Period Binomial Model for Pricing European Call Options (Numerical Values) Today (date 0)

c = 4.11 m(0) = 0.4097 b(0) = –36.8536 S = 100

6 months (date 1) cU = 8.43 m(1)U = 0.7445 b(1)U = –75.57 US = 112.82 cD = 0 m(1)D = 0 b(1)D = 0 DS = 92.24

1 year (date 2) cUU = 17.29 U2S = 127.29 cUD = 0 UDS = 104.07 cDD = 0 D2S = 85.08

$1 becomes 1 + R = $1.0253 after six months. Strike price K = $110.

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Independent of actual probabilities. Note that the actual probabilities of the stock moving up or down do not enter the valuation procedure. As in the single-period binomial model, this is because we exactly replicate the option’s payoffs at every node in the tree on the expiration date. Hence the probabilities of reaching these nodes do not affect the option’s values.

Repeat, Repeat: Risk-Neutral Pricing (Method 2) Why crank out option values by solving linear equations when we can summon the power of algebra? Plug the algebraic values from expression (18.1) into the portfolio’s value to get the synthetic call’s value at date 1: cU = V(1)U = m(1)U S(1)U + b(1)U =

cUU − cUD UcUD − DcUU US + (U’D) US (U’D) (1 + R)

Rearrange terms (keep (1 + R) in the denominator, gather cUU and cUD and rewrite) to get (1 + R) − D U − (1 + R) c + c ( U − D ) UU ( U − D ) UD cU = (18.2) 1+R Recalling Yogi Berra, “It’s like déjà -vu, all over again!” Our pseudo-probabilities are back. As before, we write the pseudo-probability of going up as 𝜋 = [(1 + R) – D]/(U – D) and going down as (1 – 𝜋) = [U – (1 + R)]/(U – D). Now you can compactly write the binomial call OPM as cU = [𝜋cUU + (1–𝜋) cUD ] / (1 + R)

(18.3)

A TWO-PERIOD BINOMIAL MODEL

401

One can develop similar formulas for call prices at the down node cD and at today’s date c. Moreover, replacing c with p gives the formula for pricing puts! Generalize this to get a formula for pricing European options.

RESULT 18.1 A Formula for Pricing Options by Repeated Application of Risk-Neutral Pricing c = [𝜋 × cU + (1–𝜋) × cD ] / (1 + R)

(18.4)

where 𝜋 = [(1 + R) – D]/(U – D), U and D are the up and down factors, respectively, and 1 + R is the dollar return. The same formula applies for puts.

The next example uses this formula.

EXAMPLE 18.3: Pricing a Call by Repeated Application of Risk-Neutral Pricing

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Using the same data as in Examples 18.1 and 18.2, we get 𝜋 = (1.0253 − 0.9224) / (1.1282 − 0.9224) = $0.50 cU = [𝜋cUU + (1 − 𝜋) cUD ] / (1 + R) = (17.29 × 0.50) /1.0253 = $8.43 (18.5)

cD = [𝜋cUD + (1 − 𝜋) cDD ] / (1 + R) = 0 c = [𝜋cU + (1 − 𝜋) cD ] / (1 + R) = (0.50 × 8.43) /1.0253 = $4.11 which verifies today’s call price.

One-Step Valuation: Prelude to the Multiperiod Model (Method 3) For pricing European options, you can simplify further and do it all in one step. In expression (18.4), plug the expressions for cU and cD into the call value c and simplify to get c=

𝜋cU + (1 − 𝜋) cD 𝜋2 cUU + 2𝜋 (1 − 𝜋) cUD + (1 − 𝜋)2 cDD = (1 + R) (1 + R)2

(18.6)

This is the two-period binomial model. The numerator computes the expected payoff using the pseudo-probabilities and the denominator discounts the cash flows. The model’s description follows:

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As the pseudo-probability of going from S to US is 𝜋, and from US to U 2 S is also 𝜋, the pseudo-probability of going from S to U 2 S is 𝜋2 , which is the pseudo probability of reaching cUU . Similarly, the pseudo-probability of reaching UDS (which corresponds to the call price of cUD ) is 𝜋 (1 – 𝜋). As there are two such paths, multiply by 2, giving an overall probability of 2𝜋(1 – 𝜋). The stock payoff D2 S (and call value cDD ) has the pseudo-probability (1 – 𝜋)2 . These pseudo-probabilities are noted in the last column of the first figure.

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Once you have these pseudo-probabilities, compute expected payoffs by multiplying each option payoff by the pseudo-probability of reaching those nodes.

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Next, compute the overall expected payoff for the call by summing up the expected payoffs computed in the previous step.

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This gives the expected payoff at time T. Discount by the dollar return (1 + R)2 to get today’s expected payoff.

EXAMPLE 18.4: A Two-Period Binomial Call Pricing Model Using the pseudo-probabilities from Example 18.3 and the option payoffs from Example 18.1 in expression (18.6) gives today’s call price:

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c = (0.25 × 17.29 + 0.50 × 0 + 0.25 × 0) /1.02532 = $4.11 One can also price European puts by any one of these three methods.1 Moreover, if you know the value of one option, you can price the other using put–call parity. 1

For example, to replicate a put, set up a portfolio V ≡ mS + b, as before. Working backward with the second-period put prices pUU = 0, pUD = $5.93, and pDD = $24.92, one gets pU = $2.89 and pD = $15.05 and, ultimately, today’s put price p = $8.75. Methods 2 and 3 work similarly.

EXAMPLE 18.5: European Put Pricing by Put–Call Parity Using the BOP data and the call price from the previous example, put–call parity for European options (Result 16.1) gives the put price as p = c + KB – S = 4.11 + 110 × (1/1.0513) –100 = $8.75

THE MULTIPERIOD BINOMIAL OPTION PRICING MODEL

18.4

The Multiperiod Binomial Option Pricing Model

We can utilize the Binomial Theorem from mathematics to develop a generalized version of the option pricing model. Many distinguished scholars have worked on this theorem including Sir Isaac Newton, truly the first rocket scientist!

Binomial Coefficients and Pseudo-probabilities The binomial coefficient (see the binomial theorem in Appendix A) provides a general formula for computing the number of ways of reaching a final payoff in the n multiperiod stock price tree. The binomial coefficient (read “n choose j”)gives (j) the number of combinations of selecting j objects out of n objects: n n! = ( j ) j! (n − j)!

(18.7a)

where the expression n! (read n-factorial) is defined as

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n! = n × (n – 1) × (n – 2) × … × 2 × 1 and 0! = 1

(18.7b)

Because the number of up movements uniquely identifies the number of different paths reaching a particular end-period stock payoff (which automatically fixes the number of down movements), we can use the binomial coefficient to compute the number of such paths. Suppose the stock price at expiration is Uj Dn−j S. As this has j up movements and (n – j) down movements, the number of paths is given by n the binomial coefficient . To compute the total pseudo-probability of reaching (j) j n−j U D S, multiply the number of ways to reach this node by the probability of each path occurring 𝜋j (1 − 𝜋)n−j . The product is given by n! 𝜋j (1 − 𝜋)n − j ( j! (n − j) ! )

(18.8)

Next, we use this to dress up our two-period model in this binomial garb.

Recasting the Two-Period Example in the Multiperiod Framework In the two-period binomial tree (n = 2), the stock price reaches U2 S or UDS or D2 S at maturity. The topmost node U2 S is reached by two ups in the stock price. Using n = 2 and j = 2, we write the pseudo-probability of the outcome as

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2 2 2! 𝜋 (1 − 𝜋)0 = 𝜋2 = 𝜋2 (2) 2! (2 − 2)!

(18.9a)

(Note that [1 – 𝜋]0 = 1.) One reaches UDS in two ways: “up and then down” and “down and then up.” Using n = 2 and j = 1, the total pseudo-probability of reaching UDS is 2 1 2! 𝜋 (1 − 𝜋) = 2𝜋 (1 − 𝜋) 𝜋 (1 − 𝜋)2 − 1 = (1) 1! (2 − 2)!

(18.9b)

Two downs in a two-period tree is reached by a unique path. Using n = 2 and j = 0, the total pseudo-probability of reaching D2 S is 2 0 2! 𝜋 (1 − 𝜋)2 = (1 − 𝜋)2 = (1 − 𝜋)2 (0) ( 0! (2 − 0)! )

(18.9c)

One can compactly express expressions (18.9a) – (18.9c) as

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2 j 2! 𝜋 (1 − 𝜋)2 − j = 𝜋j (1 − 𝜋)2 − j (j) ( j! (2 − j)! )

(18.9d)

where j = 2, 1, and 0, respectively These match the pseudo-probabilities determined earlier (for convenience, we also report these values in Figure 18.4). Recall that our two-period call valuation formula (18.6) was 𝜋2 cUU + 2𝜋 (1 − 𝜋) cUD + (1 − 𝜋)2 cDD c= (1 + R)2 Using expression (18.9a), rewrite the first expression in the numerator as 𝜋2 cUU =

2 𝜋2 (1 − 𝜋)0 × max [U2 S − K, 0] (2)

Notice that one can write the call payoff as max[0, U j D2 − j S − K] with j = 2 in this particular instance. Similarly, rewriting the other two terms in the numerator and using the summation notation yields a compact expression for the call’s price: 2

c=

2 j 1 𝜋 (1 − 𝜋)2 − j × max [0, U j D2 − j S − K] ( ) j (1 + R)2 ∑ j=0

(18.10)

THE MULTIPERIOD BINOMIAL OPTION PRICING MODEL

FIGURE 18.4: Two-Period Binomial Model for Pricing European Options Stock t=0

Stock t=1

Stock t=2 U2S

π

Call

cUU = max (U2S – K, 0)

US UDS cUD = max (UDS – K, 0)

S (1 – π)

Pseudo Probability

2 2 π (1 – π)0 = π2 2 2 1 π (1 – π)2 – 1 = 2π(1 – π) 1

DS D2S

cDD = max (D2S – K, 0)

2 0 π (1 – π)2 – 0 = (1 – π)2 0

Money Market Account (mma) Date 0 1

Date 1

Date 2

1+R

(1 + R)2

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The n-Period Binomial Option Pricing Model Replace 2 by n in expression (18.10) to get the general multiperiod binomial model, which is the main result of this chapter.

RESULT 18.2 The n-Period Binomial Option Pricing Model n

c=

n j 1 𝜋 (1 − 𝜋)n − j × max [0, U j Dn − j S − K] ( ) j (1 + R)n ∑ j=0

(18.11)

where n is the number of time periods, T is the option’s maturity date, Δt ≡ T/n is the length of each time period, 1+R ≡ erΔt is the dollar return from the money market account for a time period, j is the numbers of up movements n in n periods (thus [n – j] is the number of down movements), = (j) n! is the number of ways of choosing j “ups” out of n movements, 𝜋 j! (n − j)!

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is the pseudo-probability for the up movement ([1 – 𝜋] is the same for the down movement), U is the up factor for the stock price movement and D is the down factor, S is the initial stock price, and K is the strike price. Replacing the call payoff by max(0, K − Uj Dn−j S) gives the formula for pricing a European put option.

Who but its creator would appreciate such a hideous-looking construct? In its defense, expression (18.11): (1) is a generalization of the simpler formulas introduced earlier, (2) it just uses high school algebra (of course, with some notations), and (3) it is easily programmed into a spreadsheet program such as Microsoft Excel (see the example at the chapter’s end). To reiterate, expression (18.11) generalizes our simple formula for European call pricing developed in the previous section. The first expression within parentheses uses dollar returns to discount the cash flows. Next comes the summation sign, which adds up the “option payoffs” multiplied by the “pseudo-probability of reaching those payoffs” for all possible cases. The result is the option’s price! As in the last chapter, we can rewrite Result 18.2 more abstractly to get an expression similar to the one used in the context of the BSM model (which is explained in the next chapter):

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c = E𝜋 [c (T)] e– r T

(8.12)

where E𝜋 (⋅) denotes computing the expectation under the pseudo-probability 𝜋. Note, as emphasized before, that the option price does not depend on the actual probabilities of the stock price evolution but only on the pseudo-probabilities. Extension 18.1 shows how to choose the up and dow